A Relativistic Conical Function and its Whittaker Limits

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 101, 54 pages A Relativistic Conical F unction and its Whittak er Limits ⋆ Simon R UIJSENAARS Scho ol of Mathematics, U ni v ersity of L e e ds, L e e ds LS2 9JT, UK E-mail: siru@maths.le e ds.ac.uk URL: http://www .maths.l eeds.ac.uk/ ~ siru/ Received April 30, 2011, in f inal form Octob er 23, 2011; Published online Novem ber 0 1, 2 011 ht tp://dx.doi.or g/10 .3842/ SIGMA.2011.101 Abstract. In previous w ork we in tro duced and studied a function R ( a + , a − , c ; v , ˆ v ) that is a generalization of the hyperg eometric function 2 F 1 and the Ask ey–Wilson polyno mials. When the coupling vector c ∈ C 4 is sp ecialized to ( b, 0 , 0 , 0), b ∈ C , w e obtain a func- tion R ( a + , a − , b ; v , 2 ˆ v ) that generalizes the conica l function specializa tion of 2 F 1 and the q -Gegenbauer po lynomials. The function R is the joint eigenfunction of four analy tic dif fe- rence op erator s asso ciated with the relativistic Caloge r o–Mos e r system of A 1 t yp e, wher eas the function R corresp o nds to B C 1 , and is the joint eigenfunction of four h yp erb olic Ask ey– Wilson t ype dif fer ence op erator s. W e s how tha t the R -function admits f ive no vel int egral representations that inv olve only four h yper b olic ga mma functions and plane w av es. T aking their nonrela tivistic limit, we arrive at four r epresentations of the conical function. W e also show that a limit pro ce dure leads to tw o commuting relativistic T oda Hamiltonians and t wo comm uting dual T o da Ha milto nia ns, and that a simila rity transform of the function R conv erges to a joint eigenfunction o f the latter four dif fer ence op erato r s. Key wor ds: relativistic Calogero– Moser system; relativistic T oda system; relativistic conical function; relativistic Whittaker function 2010 Mathematics Subje ct Classific ation: 33C05; 33E30 ; 39A10 ; 81Q0 5; 81Q80 Con te n ts 1 In tro duction 2 2 The R -function as a sp ecial case of the R -f unction 6 2.1 The fu nctions R and R r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The fu nctions E and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The identitie s (2.27) and their consequences . . . . . . . . . . . . . . . . . . . . . 13 2.4 P arameter sh ifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Fiv e minimal represen tations of the R -function 17 4 Sp ecializations and nonrelativistic limit 23 4.1 Elemen tary sp ecial cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 The nonr elativistic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 ⋆ This pap er is a contribution to th e Sp ecial Issue “Relationship of Orthogonal Polynomials and Sp e- cial F unctions with Quantum Groups an d Integrable Sy stems”. The full collection is a v aila ble at http://w ww.emis.de/j ournals/SIGMA/OPSF.html 2 S. Ruijsenaars 5 The relativistic T oda case 31 5.1 T aking the relativistic T o da limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Asymptotic and analytic prop erties of F T ( η ; x, y ) . . . . . . . . . . . . . . . . . . 35 5.3 Join t eige nfunction prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6 The nonrelativistic T o da case 40 A The hyperb olic gamma function 43 B Pro of of (A.33 ) 46 C F ourier transform form ulas 48 References 53 1 In tro duction This article ma y b e view ed as a contin u ation of our previous work on a ‘relativistic ’ generaliza- tion R of the Gauss hyp ergeometric fun ction 2 F 1 , in trodu ced in [1]. Th e latter pap er and t wo later parts in a series [2, 3] w ill b e referred to as I, I I and I I I in the sequel. The def inition of the R -fun ction in I is in terms of a con tour in tegral, whose integrand inv olve s eigh t h yp erb olic gamma f unctions. (W e review this in Section 2, cf. (2.1)–(2.5).) In recen t y ears, v an de Bu lt [4] tied in the R -fun ction with the n otion of m o dular double of the quantum group U q ( sl (2 , C )), as d ef ined by F addeev [5]. As a spin-of f, he obtained a new represent ation of the R -fun ction. Also, v an de Bu lt, Rains and Stokman [6 ] ha v e shown (among other things) that the 8-v ariable R -fun ction R ( a + , a − , c ; v , ˆ v ) , a + , a − , v , ˆ v ∈ C , a + /a − / ∈ ( −∞ , 0] , c ∈ C 4 , (1.1) can b e obtained as a limit of Spiridono v’s 9-v ariable hyperb olic hyp ergeometric fu nction [7]. Their no v el viewp oin t leads to a third repr esen tation for the R -fu n ction. (See Prop osition 4.20 and Theorem 4.21 in [6] for the latter t wo r ep resen tations.) In this p ap er we are concerned with a 5-v ariable sp ecializatio n of the R -function, def in ed by R ( a + , a − , b ; x, y ) ≡ R ( a + , a − , ( b, 0 , 0 , 0); x, y / 2) . (1.2) Suitable discretiza tions of this function give rise to the q -Gegen bauer p olynomials, whereas discretizations of the R -function yield the Askey–Wi lson p olynomials, cf. I; moreo ver, the non- relativistic limit of the R -function yields the conical function sp ecialization of 2 F 1 . Hence it ma y b e viewed as corresp onding to the Lie alg ebra A 1 , whereas the R -function can b e tie d in with B C 1 . The k ey new result of this pap er concerning R consists of the in teg ral representa tion R ( b ; x, y ) = r α 2 π G (2 ib − ia ) G ( ib − ia ) 2 Z R dz G ( z + ( x − y ) / 2 − ib/ 2) G ( z − ( x − y ) / 2 − ib/ 2) G ( z + ( x + y ) / 2 + ib/ 2) G ( z − ( x + y ) / 2 + ib/ 2) . (1.3) Here and thr oughout th e p ap er w e use parameters α ≡ 2 π /a + a − , a ≡ ( a + + a − ) / 2 , (1.4) G ( a + , a − ; z ) is the h yp erb olic gamma fu n ction (cf. App endix A), and the dep endence on a + , a − is suppressed. (W e shall often do this wh en no confusion can arise.) F urthermore, in (1.3) we c ho ose at f irst ( a + , a − , b, x, y ) ∈ (0 , ∞ ) 2 × (0 , 2 a ) × R 2 . (1.5) A Relativi stic Conical F unction and its Whittak er Limits 3 By con trast to the previous th r ee int egral repr esen tations follo wing fr om I, [4] and [6], the in tegrand in (1.3) inv ol v es only four hyp er b olic gamma functions. W e also obtain several closel y related repr esentati ons that inv olve in add ition plane wa ve s, cf. (3.47 )–(3.51). As will transp ire in Section 3, up on u s ing the f irst one (1.3) of these nov el represen tations (whic h w e dub ‘minimal’ represent ations) to in trodu ce the R -function, it is p ossib le to rederive in a more transparent and self-con tained wa y a g reat many feat ures th at also follo w up on sp ecializat ion of the R - function th eory , d ev eloped not on ly in I, I I and I I I, but also in our later p ap ers [8] and [9]. Moreo v er, sp ecial cases and limits of the R -fu nction are far more easily obtained from the minimal r epresen tations than f r om the original integral representat ion of I or from the alternativ e represent ations follo wing fr om [4] and [6]. (The in tegrands of these earlier rep resen tations inv olv e at lea st eigh t h yp erb olic gamma function facto rs.) A surve y of the resu lts of I–I I I and [8 ] can b e found in [10], bu t the def inition (1.2) of the A 1 -analog of the ( B C 1 ) R -fu nction dates bac k to the more recen t p ap er [9]. In S ection 2, we review in p articular the p ertinen t results fr om [9]. Ho wev er, we ha v e o ccasion to ad d a lot more information that follo w s by sp ecializing pr evious f indin gs concerning the R -fun ction and related functions to their A 1 coun terparts. T his includ es the asymptotic b eha vior and Hilb ert s p ace prop erties obtained in I I and I I I, resp., whic h are adapted to the A 1 setting in Sub s ection 2.2, and the p arameter s h ifts ob tained in [8], whic h we fo cus on in Sub section 2.4. Moreo v er, in (2.27) w e detail the connection of the renormalized function R r ( a + , a − , b ; x, y ) ≡ G ( ib − ia ) G (2 ib − ia ) R ( a + , a − , b ; x, y ) , (1.6) to the A 1 t yp e functions M ( ma + + na − ; x, y ), m, n ∈ Z , whic h featured in our previous pa- p ers [11 ] and [12]. W e present the pro of of ( 2.27) in Su bsection 2.3, together w ith v arious corollaries. Altoget her, Sectio n 2 inv ok es a considerable amoun t of information from our previous w ork. W e ha v e attempted to sketc h this in s uc h a wa y that the reader need only consult the p ertinent pap ers for quite tec hnical asp ects (in case of doub t and /or inclinati on, of cours e). Ev en so, it is pr obably advisable to skim through Section 2 at f irs t reading, referring bac k to it w hen the need arises. By con trast, S ection 3 (combined with App en dices A and C) is largely self-con tained. Its starting p oin t is a h yp erb olic functional identit y that f irst arose as a sp ecializa tion of elliptic functional id en tities expr essing the relation of certain Hilb ert–Sc hmidt integral k ernels to the elliptic B C 1 relativistic Calogero–M oser dif ference op erators int ro du ced by v an Diejen [13]. W e need not inv oke these iden tities (which can b e found in [14], cf. also [15]), since the relev ant h yp erb olic v ersion is qu ite easily prov ed dir ectly . The ke y p oint is that the h yp erb olic identi ties can b e rewritten in term s of t w o pairs of hyp erb olic A 1 -t yp e relativistic Calogero– Moser dif fe- rence op erators A ± ( b j ; x ), j = 1 , 2, w ith distinct couplings b 1 , b 2 . Th e dif ference op erators are giv en by A δ ( b ; x ) ≡ s δ ( x − ib ) s δ ( x ) T x ia − δ + s δ ( x + ib ) s δ ( x ) T x − ia − δ , δ = + , − . (1.7) Here, the trans lations are def ined on analytic fun ctions by ( T z c f )( z ) ≡ f ( z − c ) , c ∈ C ∗ . (1 .8) Also, th roughout this pap er we use the abb reviations s δ ( z ) = sinh( π z /a δ ) , c δ ( z ) = cosh( π z /a δ ) , e δ ( z ) = exp( π z /a δ ) , δ = + , − . (1.9) 4 S. Ruijsenaars Cho osing b 1 = b , b 2 = 0, an auxiliary f unction B ( b ; x, y ) can b e def in ed th at satisf ies the eigen v alue equations (at f irst u nder certain restrictions on the B -argumen ts) A δ ( b ; x ) B ( b ; x, y ) = 2 c δ ( y ) B ( b ; x, y ) , δ = + , − . (1.10) More sp ecif ically , the function B ( b ; x, y ) is the F ourier transform of the hyp erb olic k ernel func- tion, whic h is a pro du ct of f our h yp erb olic gamma fu nctions. When we w r ite the integrand of the in tegral d ef ining B as a pro duct of tw o factors th at inv olve only t w o hyp erb olic gamma fun ctions, we can u se the Planc herel relation an d th e explicit F our ier transform form ula for factors of this t yp e (deriv ed in App end ix C) to obtain t w o new in tegral represent ations for B . In p articular, this leads to a f u nction C ( b ; x, y ) giv en by C ( b ; x, y ) ≡ r α 2 π Z R dz G ( z + ( x − y ) / 2 − ib/ 2) G ( z − ( x − y ) / 2 − ib/ 2) G ( z + ( x + y ) / 2 + ib/ 2) G ( z − ( x + y ) / 2 + ib/ 2) . (1.11) Comparing (1.11), (1.3) and (1.6), w e read of f R r ( b ; x, y ) = G ( ib − ia ) C ( b ; x, y ) . (1.12) Ho w ev er, w e n eed a further study of the C -fu nction (1.11) to arrive at a pro of of this relation to the fu nction R r , as def in ed originally by (1.6) and (1.2). In deed, as already alluded to b elo w (1.5 ), we can use (1.11 ) as a starting p oin t to derive m an y features that C and R r ha v e in common. In p articular, the general analysis in Ap p end ix B of I can b e applied to the integral on the r .h .s. of (1.11), which yields a complete elucidation of the b eh avior of C ( b ; x, y ) un der meromorphic con tin uation. Moreo v er, via the A∆Es (analytic dif ference equations) (1.10) and the m anifest in v ariance of C u nder in terc h anging x and y , it follo ws that C ( b ; x, y ) is a joint eigenfunction of th e four A∆Os (analytic dif ference op erators) A + ( b ; x ) , A − ( b ; x ) , A + ( b ; y ) , A − ( b ; y ) , (1.13) with eige n v alues 2 c + ( y ) , 2 c − ( y ) , 2 c + ( x ) , 2 c − ( x ) , (1.14) resp. This is also the case for R r ( b ; x, y ) and, moreo v er, the equalit y (1.12) can b e sho wn for the sp ecial case y = ib by a further app lication of App endix C. The general case then follo ws b y a un iqu eness argument already used in Subsection 2.3. W e reconsider the s p ecial b -v alues b mn ≡ ma + + na − , m, n ∈ Z , (1.15) in Su bsection 4.1, inasmuc h as they satisfy b mn ∈ (0 , 2 a ). Indeed, the new F ourier transform represent ations in Sectio n 3 are only w ell def ined for b ∈ (0 , 2 a ), bu t they can b e explicitly ev aluated by a residue calculatio n when b is of this form. The key p oint is that the G -ratios in the integrand can then b e wr itten in terms of the h yp erb olic cosines c ± ( z ) by usin g the G - A∆Es (A.2). In principle, this yields again the functions M ( b mn ; x, y ) from [11], but w e ha v e not tried to push through a direct equalit y pro of (as opp osed to app ealing to u niqueness). Subsection 4.2 deals w ith the nonr elativistic limit. Sp ecializing th e results of I yields the h yp ergeometric fun ction in terms of whic h the conical function can b e expressed (cf. C hapter 14 in [16 ]). The f iv e m inimal repr esentati ons (3.47)–(3.51) of the R -function lead to four repre- sen tations (4.45)–(4.48) of the limit fun ction. Rewr iting them in terms of the conical function, three of these can b e found in the literature (by lo oking rather hard). This is reassurin g, since just as in I w e were not able to get rigorous con trol on the nonrelativistic limits. A Relativi stic Conical F unction and its Whittak er Limits 5 In order to describ e the results of S ection 5, w e b egin b y recalling that in our pap er [17] we arriv ed at relati vistic nonp erio d ic T o da N -particle sys tems by taking a limit of the relativistic h yp erb olic Caloge ro–Moser N -particle systems. In this limit the self-du alit y of the latter is not preserved, inasmuc h as the d ual commuting Hamiltonians hav e a very dif ferent c h aracter from the def ining Hamiltonian and its comm uting family . Sp ecialized to the pr esent con text, this limit can b e used to obtain a joint eigenfunction of t w o T o da Hamiltonians H T ± ( η ; x ) and t w o dual T o d a Hamiltonians ˆ H T ± ( η ; y ), with the real parameter η pla ying th e role of a coupling constan t. The limit transition pro ceeds in t w o stage s. The f irst step is to set b = a − iγ , γ ∈ R . (1.16) A t the classical lev el the analogous b -choic e still yields real-v alued Hamiltonians with a well- def ined self-dual action-angle map an d scattering theory [18 ]. Corresp ondingly , the four reduced N = 2 quan tum Hamiltonians at issue here are still formally self-adjoin t for this b -c hoice. (They are similarit y transforms of the A∆Os (1.13) with a we igh t fun ction factor.) Moreo v er, r estricting atten tio n to ( a + , a − , x, y ) ∈ (0 , ∞ ) 2 × R 2 , (1.17) their j oint eigenfunction remains real-v alued , although this realit y prop ert y is no longer manifest: It hinges on a symmetry p r op erty u n der taking b to 2 a − b , which translates in to ev enness in the parameter γ . The next s tep is to s ubstitute x → x + Λ , γ → η + Λ , (1.18) and tak e Λ to ∞ . In this limit the Hamiltonians and their joint eigenfunction conv erge, whereas the d ual Hamiltonians must b e multiplie d b y a factor e δ ( − Λ) to obtain a f inite limit. Th is can b e understo o d from their Λ-dep endent eigen v alues 2 c δ ( x + Λ) follo w ing from th e x -sh if t (1.18), cf. (1.14). In deed, after m ultiplica tion by this renormalizing factor the eige n v alues h a v e the f inite limits e δ ( x ), δ = + , − . The f iv e representa tions of the R -function giv e r ise to four repr esentati ons of the relativistic T o da eige nfun ction F T ( η ; x, y ), namely (5.25), (5.26), (5.32) and (5.3 3). Suitably p aired of f, ho w ev er, these dif feren t form ulas express real-v aluedness with (1.17) in ef fect. T aking th is in to accoun t, we wind up with t w o essen tially dif f eren t representa tions that are intert w in ed via the Planc herel form ula for the F ourier transform. Th e k ey form ula in v olv ed here is deriv ed in Corollary C.2 . The results just delineated can b e found in Sub section 5.1. In Sub s ection 5.2 we f irst stud y the asymptotic b eha vior of F T ( η ; x, y ) f or x → ± ∞ and y → ∞ . W e then clarify the analyticit y prop erties of F T ( η ; x, y ) by introdu cing a similarit y transform H ( x − η, y ). Using the four represent ations (5.54)–(5.57) of the latter, we sho w that th e function H ( x, y ) is holomorphic for ( x, y ) ∈ C 2 . Subsection 5.3 deals w ith the joint eigenfun ction p rop erties of F T ( η ; x, y ) and its similarit y transforms. F ormally , these follo w fr om those of the R -fun ction. Ho w ev er, the T o da limit is n ot easy to cont rol analyticall y , and the dir ect deriv ation of the eigen v alue equations is not to o hard and quite illum in ating. Our results in Section 5 hav e some o v erlap w ith earlier results by Kharc hev, Leb ed ev and Semeno v-Tian-Shansky [19], who obtained f unctions closely related to F T ( η ; x, y ) from the viewp oint of harmonic analysis for F addeev’s mo dular double of a qu an tum group [5]. The nonrelativistic nonp erio dic T o da eig enfunctions are widely kno w n as Whittak er functions, and mean while it has b ecome customary to call eigenfun ctions f or q -T o da Hamiltonians Whittak er 6 S. Ruijsenaars functions as w ell. In p articular, q -Whittak er fu nctions w ere in trod uced b y Olshanetsky and Rogo v for ran k 1 (their w ork can b e traced from [20]) and b y Etingof for arbitrary rank [21], and these functions ha v e b een f urther studied in v arious later pap ers (see e.g. [22] and r eferen- ces giv en there). W e w ould lik e to stress that these functions are quite dif ferent from the ones at issue h ere an d in [19]. The crux is that the f orm er are only w ell def ined for q not on the unit circle, whereas here and in [19] the eigenfunctions ha v e a sym m etric dep endence on t w o generically distinct q ’s, giv en b y q + = exp( iπ a + /a − ) , q − = exp( iπ a − /a + ) . (1.19) This stat e of af fairs is closely related to the dif ferent c haracter of the trigonometric gamma function (more widely kno wn as the q -gamma fu nction, w ith the restriction | q | 6 = 1 b eing indis- p ensable) and the h yp erb olic gamma fu nction (which dep ends on parameters a + and a − in the righ t half plane). In Section 6 we study the non r elativistic limit of the representati ons of the r elativistic eigen- function, arriving at t w o distinct repr esen tations for the nonp erio dic T o da eigenfunction that ha v e b een kno wn for a long time. Just as for the relativistic case, its prop erty of b eing also an eigenfunction for a dual Hamiltonian seems not to ha v e b een observ ed b efore. (These dualit y features are the quantum coun terparts of duality features of th e p ertinent actio n-angle maps, f irst p oin ted out in [17].) T o control one of the t w o p ertinen t limits, a no v el limit transition for the h yp erb olic gamma function is needed, whose pr o of is r elegate d to App end ix B. 2 The R - function as a sp ecial case of the R -function 2.1 The functions R and R r The R -fu nction (1.1) is def in ed as a contour in tegral o v er a v ariable z , with th e z -dep end ence of the integ rand enco ded in a pro d u ct of eigh t h yp erb olic gamma fun ction factors. (See App endix A for a review of the relev ant features of the h yp erb olic gamma function.) Sp ecif ically , w ith suitable restrictions on the eigh t v ariables, the R -function is giv en b y R ( c ; v , ˆ v ) = r α 2 π Z C F ( c 0 ; v , z ) K ( c ; z ) F ( ˆ c 0 ; ˆ v , z ) dz . (2.1) Here w e hav e ˆ c 0 ≡ ( c 0 + c 1 + c 2 + c 3 ) / 2 , (2.2) F ( d ; y , z ) ≡ G ( z ± y + id − ia ) G ( ± y + id − ia ) , (2.3) (with f ( w ± y ) denoting f ( w + y ) f ( w − y )), and K is giv en by K ( c ; z ) ≡ 1 G ( z + ia ) 3 Y j =1 G ( is j ) G ( z + is j ) , (2.4) with new parameters s 1 ≡ c 0 + c 1 − a − / 2 , s 2 ≡ c 0 + c 2 − a + / 2 , s 3 ≡ c 0 + c 3 . (2.5) Also, r ecall a and α are def in ed by (1.4). W e do not need the def inition of the con to ur C for general v ariable c hoices (this is d iscussed in I and Section 4 of the survey [10]); ins tead w e presently def ine C for the cases at issue. F or the sp ecial c -c hoice in (1.2) w e can use the d u plication form ula (A.10) to obtain K (( b, 0 , 0 , 0); z ) = K ( b ; z ) , (2.6) A Relativi stic Conical F unction and its Whittak er Limits 7 where K ( b ; z ) ≡ 1 G ( z + ia ) G (2 ib − ia ) G ( ib − ia ) G ( z + ib − ia ) G (2 z + 2 ib − ia ) . (2.7) Using also the ref lection equation (A.6) we dedu ce that R is give n b y R ( b ; x, y ) =  α 2 π  1 / 2 G (2 ib − ia ) G ( ib − ia ) G ( ± x − ib + ia ) G ( ± y / 2 − ib/ 2 + ia ) × Z C G ( z ± x + ib − ia ) G ( z + ib − ia ) G ( z ± y / 2 + ib/ 2 − ia ) G ( z + ia ) G (2 z + 2 ib − ia ) dz . (2.8) F or the v ariable c hoic e that is most relev ant for Hilb ert space purp oses, namely , a + , a − , b, x, y > 0 , (2.9) the cont our C ma y b e c hosen equal to the r eal line in the z -plane, indented do wn w ards near 0 so as to av oid a p ole of K ( b ; z ). F rom (A.17)–(A.1 8) it f ollo ws that the p oles of K ( b ; z ) are lo cated on th e imaginary axis at z − z k l = 0 , z − z k l + ib = ia + / 2 , ia − / 2 , 0 , k , l ∈ N . (2.10) Th us th ey are ab o v e the con tour, whereas th e remaining z -p oles of the integ rand at z + z k l = ± x − ib, ± y / 2 − ib/ 2 , k , l ∈ N , (2.11) are b elo w C . F rom the abov e represen tation it is immediate that R is symmetric und er the interc h ange of the parameters a + and a − : R ( a + , a − , b ; x, y ) = R ( a − , a + , b ; x, y ) . (2.12) It is not at all clear, ho w ev er, that R is also symmetric un der the inte rc hange of the p ositions x and y : R ( a + , a − , b ; x, y ) = R ( a + , a − , b ; y , x ) . (2.13) This self-du alit y feature follo ws in particular from a second r elation b et w een R and R , namely , R ( b ; x, y ) = R (( b, b, b, b ) / 2; x/ 2 , y ) . (2.14) (This is equation (4.8) in [9].) Indeed, this second c -c hoice yields the same function K ( b ; z ) as the f irst one, so that sub stitution of (2.1 ) (with the same con tour C ) now yields (2.8) with x and y inte rc hanged on the r.h.s. There are t w o more c -choi ces that lead from R to R , namely , ( b, 0 , b, 0) and ( b, b, 0 , 0). Sp ecif ically , from equations (4.6) and (4.7) in [9 ] we ha v e R ( a + , a − , b ; x, y ) = R ( a + , 2 a − , ( b, 0 , b, 0); x, y ) , (2.15) R ( a + , a − , b ; x, y ) = R (2 a − , a + , ( b, b, 0 , 0); x, y ) . (2.16) F rom the def inition of the R -function we then obtain alternativ e in teg ral represen tations for the R -fun ction from whic h the self-dualit y pr op ert y (2.13) is m an if est. (In deed, since w e hav e c 0 = ˆ c 0 = b for th ese tw o c hoice s, the in tegrand is inv ariant u nder the in terc hange of x and y .) On the other hand , the mo dular in v ariance pr op erty (2.12) is not at all clear, sin ce the integ ral represent ations inv ol v e the h yp erb olic gamma function with a − replaced by 2 a − . Using (A.11), 8 S. Ruijsenaars they can b e re-expressed in terms of the mo dular inv ariant function G ( a + , a − ; z ). Ho w ev er , the resulting int egrand is then still not mo d ular in v arian t. Since it seems not to simplify and d o es not look illuminating, w e do not detail it any fur ther. The analyticit y p rop erties of the R -function are kno wn in great detail from Theorem 2.2 in I, cf. also Section 4 in the s urve y [10]. Com bining this theorem with the def in ition (1.2) of R and its self-dualit y prop ert y (2.13), w e dedu ce in particular that R extends fr om th e in terv als (2.9) to a m eromorphic f u nction in b , x and y , whose p oles in x and y can on ly o ccur at the lo cations ± z = 2 ia − ib + z k l , z = x, y , k , l ∈ N . (2.17) W e pro ceed to list fur th er consequences of th e R -fu nction theory for R . Tw o features that are clea r from eac h of th e ab o v e integral represen tations are ev enness and scale in v ariance (giv en scale in v ariance of G ): R ( b ; x, y ) = R ( b ; δ x , δ ′ y ) , δ, δ ′ = + , − , (2.18) R ( a + , a − , b ; x, y ) = R ( λa + , λa − , λb ; λx, λy ) , λ > 0 . (2.19) A less obvio us feature is the explicit ev aluation R ( b ; x, ib ) = 1 . (2.20) It follo ws fr om th e form ula R ( c ; v , i ˆ c 0 ) = 1 , (2.21) (cf. equation (3.26) in I or Section 6 in [10]), by using an y of the four relations (1.2), (2.1 4), (2.15), (2.1 6). Def ining next R n ( x ) ≡ R ( b ; x, y n ) , y n ≡ ib + ina − , n ∈ N , (2.22) the eigen v alue A∆E (analytic dif ference equation) for A + ( b ; y ) enta ils s + ( y n − ib ) s + ( y n ) R n − 1 ( x ) + s + ( y n + ib ) s + ( y n ) R n +1 ( x ) = 2 c + ( x ) R n ( x ) . (2.23) In view of (2.20), it follo ws fr om th is that R n is of the form R n ( x ) = P n ( c + ( x )) , (2.24) where P n ( z ) is a p olynomial in z of degree n and p arit y ( − ) n . The relation of th ese p olynomials to the q -Gegen bauer p olynomials and to the Ask ey– Wilson p olynomials associated with the f our relev ant c -c hoices is detailed at the end of Section 4 of [9]. The renormalized R -fu nction R r giv en by (1.6 ) is the counterpart of the renormalized R - function R r obtained from (2.1) by omitting the p ro du ct Q j G ( is j ) in K , cf. (2.4). (T o see this, use (A.6) and (A.10).) Clearly , it shares th e features (2.12), (2.13), (2.18) and (2.19 ) of the R -function, whereas (2.20 ) is replaced b y R r ( b ; x, ib ) = G ( ib − ia ) G (2 ib − ia ) . (2.25) The renormalizing factor in the fun ction R r ensures that it has n o p oles that are indep endent of x and y , cf. Theorem 2.2 in I. More precisely , R r ( a + , a − , b ; x, y ) extends to a fu nction that is meromorphic in the domain D + ≡  ( a + , a − , b, x, y ) ∈ C 5 | Re a + > 0 , Re a − > 0  , (2.26) and whose p oles can only o ccur at the lo cations (2.17) . A Relativi stic Conical F unction and its Whittak er Limits 9 It is not ob vious, but true that for the sp ecial b -c hoices b mn (1.15) w e h a v e an equalit y R r ( b mn ; x, y ) = M ( b mn ; x, y ) + M ( b mn ; − x, y ) , (2. 27) where M ( b mn ; x, y ) is th e function def ined at the end of Section I I I in our pap er [11]. Therefore, R r ( b ; x, y ) is the contin u ous (indeed, real -analytic) in terp olation to arbitrary b ∈ R of the function giv en by equation (3.74) in [11], wh ic h is d ef ined only for the b -v alues b mn . (Note the latter are d ense in R when th e ratio a + /a − is ir r ational.) It will not cause surprise that in the f r ee case we ha v e M ( b 00 ; x, y ) = exp( iαxy / 2) . (2.28) It w ould tak e us too far af ield, ho w ev er, to detail all of the functions M ( b mn ; x, y ), m, n ∈ Z . F or our p urp oses it is enough to sp ecify their general structure: They are ‘elemen ta ry’ in the sense th at they can b e written M ( b mn ; x, y ) = exp( iαxy / 2) R mn ( e + ( x ) , e − ( x ) , e + ( y ) , e − ( y )) , (2.29) where R mn is a rational fun ction of its four arguments, cf. S ection I I I in [11]. In S ubsection 2.4 w e deduce this structur e in another wa y (namely , by exploiting parameter shifts). Moreo v er, for the case wh ere m and n are not b oth p ositiv e or b oth non-p ositiv e, this structure can b e understo o d fr om the no v el F ourier transform represen tatio ns (3.48)–(3.51), cf. S ubsection 4.1. W e p ostp one the pro of of the equality assertion (2.27) to Subsection 2.3. An ingred ient of this pro of is the asymptotic b ehavior of R r ( b ; x, y ) as x go es to ∞ , and this is most easily obtained as a corollary of the asymptotics of a closely related function E( b ; x , y ), d ef ined b y (2.38). 2.2 The functions E and F The function E( b ; x, y ) can b e viewed as a sp ecializa tion of the fu nction d enoted E ( γ ; v , ˆ v ) in I I and [10]. T he relati on b et w een γ and c reads γ ( a + , a − , c ) = ( c 0 − a, c 1 − a − / 2 , c 2 − a + / 2 , c 3 ) . (2.30) In particular, th e ‘fr ee’ case c = 0 yields γ f ≡ ( − a, − a − / 2 , − a + / 2 , 0) . (2.31) The sw itc h from c to γ is crucial for unco v erin g f urther symmetries: The fu nction E ( γ ; v , ˆ v ) is inv arian t under D 4 transformations on γ (i.e., p erm utations and ev en sign c hanges), cf. I I. (In [6] this D 4 symmetry has b een reobtained in a qu ite dif feren t wa y .) It is def ined by E ( γ ; v , ˆ v ) = χ ( γ ) c ( γ ; v ) c ( ˆ γ ; ˆ v ) R r ( γ ; v , ˆ v ) . (2.32) Here, the generalized ( B C 1 ) Harish-Chandra c -fun ction is giv en b y c ( γ ; v ) ≡ 1 G (2 v + ia ) 3 Y µ =0 G ( v − iγ µ ) , (2.33) the dual of γ by ˆ γ ≡ J γ , J ≡ 1 2     1 1 1 1 1 1 − 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1     , (2.34) 10 S. Ruijsenaars and the constan t b y χ ( γ ) ≡ exp  iα [ γ · γ / 4 −  a 2 + + a 2 − + a + a −  / 8]  , α = 2 π /a + a − . (2.35) Denoting the γ -vec tors corresp onding to the t w o one-parameter families c = ( b, 0 , 0 , 0) , ( b, b, b, b ) / 2 , (2.36) b y γ (1) , γ (2) , it is easy to chec k that J γ (1) = γ (2) . (2.37) Def in ing E( b ; x, y ) ≡ E  γ (1) ; x, y / 2  , (2.38) a straigh tfo rwa rd calculation (using the dup lication formula (A.1 0)) yields E( b ; x, y ) = φ ( b ) c ( b ; x ) c ( b ; y ) R r ( b ; x, y ) , (2.39) where w e hav e introdu ced the constan t φ ( b ) ≡ exp( iαb ( b − 2 a ) / 4) , (2.40) and generaliz ed ( A 1 ) Harish-Ch andra c -function c ( b ; z ) ≡ G ( z + ia − ib ) G ( z + ia ) . (2.41) Recalling (2.14) and u sing (2.37), it readily follo ws that we also ha v e E( b ; x, y ) = E  γ (2) ; x/ 2 , y  . (2.42) It in v olv es more work to obtain the relations b etw een E and E corresp onding to (2.15) and (2.1 6). Setting γ (3) ≡ γ ( a + , 2 a − , ( b, 0 , b, 0)) , γ (4) ≡ γ (2 a − , a + , ( b, b, 0 , 0)) , (2.43) these are give n b y E( a + , a − , b ; x, y ) = E  a + , 2 a − , γ (3) ; x, y  = E  2 a − , a + , γ (4) ; x, y  . (2.44) (These f orm ulas amount to s p ecial cases of the d oubling identit y for the E -function obtained in Section 6 of [9].) The relation (2.3 9) b et w een E and R r yields a similarity transformation turning the A∆Os (1.13) in to A ± ( b ; x ), A ± ( b ; y ), where A δ ( b ; z ) ≡ c ( b ; z ) − 1 A δ ( b ; z ) c ( b ; z ) = T z ia − δ + V δ ( b ; z ) T z − ia − δ , δ = + , − , (2.45) V δ ( b ; z ) ≡ s δ ( z + ib ) s δ ( z − ib + ia − δ ) s δ ( z ) s δ ( z + ia − δ ) . (2.46) F rom this it is easy to verify th at these A∆Os are f ormally self-adjoin t operators on L 2 ( R ) (by con trast to the A∆Os (1.1 3) ), and that they are inv arian t under the transform ation b 7→ a + + a − − b. (2.47) A Relativi stic Conical F unction and its Whittak er Limits 11 It is not ob vious, but true that w e also hav e E( b ; x, y ) = E( a + + a − − b ; x, y ) . (2.48) This s y m metry pr op ert y can b e derived from (2.38) and the D 4 in v ariance of the E -fun ction: W e ha v e γ (1) = ( b − a, − a − / 2 , − a + / 2 , 0) , (2.49) so the map b 7→ 2 a − b yields a γ -v ector related to γ (1) b y a sign f lip of the f ir s t and last comp onent . Com bining (2.39) and (2.41) w ith th e analyticit y features of R (cf. the paragraph con- taining (2.17)), we deduce that E( b ; x, y ) is m eromorphic in b , x and y , with b -indep en d en t p ole lo cations z = − 2 ia − z k l , z = x, y , k, l ∈ N , (2.50) corresp ondin g to the fact or G ( x + ia ) G ( y + ia ), and b -dep endent p oles at z = ib + z k l , z = − ib + 2 ia + z k l , z = x, y , k , l ∈ N . (2.51) The main disadv antag e of the function E ( b ; x, y ) compared to the R r -function is that it is not ev en in x and y , since the c -functions in (2.39) are not ev en. Instead, it satisf ies E( b ; − x, y ) = − u ( b ; x )E( b ; x , y ) , (2.52) where u ( b ; z ) ≡ − c ( b ; z ) /c ( b ; − z ) = − G ( z ± ( ia − ib )) /G ( z ± ia ) . (2.53) On the other h and, E in herits all other imp ortan t prop erties of R r , and is the simplest function to use for Hilb ert space purp oses. In particular, it has the ‘unitary asymptotics’ E( b ; x, y ) ∼ exp( iαxy / 2) − u ( b ; − y ) exp ( − iαxy / 2) , (2.54) b ∈ R , y ∈ (0 , ∞ ) , x → ∞ , cf. Theorem 1.2 in I I. Here, the u -function encod es the scattering asso ciated with the A∆Os A ± ( b ; x ), reinte rpreted as comm uting self-adjoint op erators on the Hilb ert space L 2 ((0 , ∞ ) , dx ). More p recisely , us ing corresp ond in g results on th e E -function f rom I I and I I I, it follo ws that the generalized F ourier transform F : C ≡ C ∞ 0 ((0 , ∞ )) ⊂ ˆ H ≡ L 2 ((0 , ∞ ) , dy ) → H ≡ L 2 ((0 , ∞ ) , dx ) , (2.55) def ined b y ( F ψ )( x ) ≡  α 4 π  1 / 2 Z ∞ 0 E( b ; x, y ) ψ ( y ) dy , ψ ∈ C , (2.56) extends to a unitary op er ator, provided the coupling b is su itably restricted. S p ecif ically , it suf f ices to require b ∈ [0 , a + + a − ] . (2.57) The self-adjointness of the op erators ˆ A ± ( b ) on H asso ciated to the A∆Os A ± ( b ; x ) for b in this in terv al can then b e easily u n dersto o d from the un itarit y of F : they are the pu llbac ks to H under F of the self-adjoin t op erators of m ultiplication by 2 c ± ( y ) on ˆ H . 12 S. Ruijsenaars As already mentioned, these statemen ts follo w f r om I I and I I I b y sp ecialization, bu t it ma y help to lo ok f irst at Section 9 in the sur v ey [10 ]. Starting from the representa tion (2.38), the v ector γ (1) b elongs to the p olytop e P giv en by equation (9.2) in [10], pro vided b ∈ (0 , 2 a ). Therefore, the transform asso ciated with E ( γ (1) ; v , ˆ v ) (def ined by equation (9.4)) is an isometry . As a consequence, the trans form (2.56) is an isometry . (Th e normalization factor in (2.56) dif fers from that in equation (9.4) in [10] to accommo date the scale factor 1/2 in the y -dep endence of E in (2.38) .) I sometry of the inv erse transform is then clear fr om the self-dualit y of E( b ; x, y ). Next, for b = 0 w e ha ve the identit y E(0; x, y ) = R r (0; x, y ) = 2 cos ( αxy / 2) , (2.58) cf. (2.3 9)–(2.41 ) and equation (7.33) in [10] (also, n ote that for b = 0 we ha v e γ (1) = γ f , cf. (2.4 9) and (2.31) ). In view of the symmetry (2.4 8), it f ollo ws that F amounts to the cosine tran s form for b = 0 and b = 2 a , so th ese transforms are un itary as w ell. More generally , we obtain a family of unitary op erators F ( a + , a − , b ) , ( a + , a − , b ) ∈ Π u ≡ (0 , ∞ ) 2 × [0 , a + + a − ] , (2.59) whic h is str on gly con tin uous on th e p arameter set Π u and satisf ies F ( a + , a − , b ) = F ( a + , a − , a + + a − − b ) , (2. 60) cf. Theorem 3.3 in II I. The G -function asymp totics (A.13 ) en tail s that the c -fu n ction (2.4 1) has asymptotics c ( b ; x ) ∼ φ ( b ) ± 1 exp( ∓ αbx/ 2) , Re ( x ) → ±∞ , (2.61) with φ ( b ) giv en by (2.40) . Hence the u -function (2.5 3) has asymptotics u ( b ; x ) ∼ − φ ( b ) ± 2 , Re ( x ) → ±∞ . (2.62) Also, th e ref lection equation (A.6) and the complex conjugation relation (A.9) entai l u ( b ; − x ) u ( b ; x ) = 1 , | u ( b ; x ) | = 1 , b, x ∈ R . (2.63) Th us, if we set F( b ; x, y ) ≡ φ ( b ) − 1 ( − u ( b ; x )) 1 / 2 ( − u ( b ; y )) 1 / 2 E( b ; x, y ) , b, x, y > 0 , (2.64) (with the squ are r o ot phase factors reducing to 1 for b = 0), then F has asymptotics F( b ; x, y ) ∼ [ − u ( b ; y )] 1 / 2 exp( iαxy / 2) + [ − u ( b ; y ] − 1 / 2 exp( − iαxy / 2) , x → ∞ , (2.65) and if we replace E b y F in the ab o v e u nitary transform (2.56), w e retain u nitarit y . In tro ducing the weigh t f unction w ( b ; x ) ≡ 1 /c ( b ; ± x ) = G ( ± x + ia ) /G ( ± x + ia − ib ) , (2.66) w e ha v e w ( b ; x ) > 0 , b, x > 0 , (2.67) and w e can also write F in terms of R r as F( b ; x, y ) = w ( b ; x ) 1 / 2 w ( b ; y ) 1 / 2 R r ( b ; x, y ) , b, x, y > 0 , (2.68) A Relativi stic Conical F unction and its Whittak er Limits 13 with the p ositiv e square ro ots u ndersto o d. Hence, F is a j oin t eigenfunction of th e A∆Os H ± ( b ; x ) and H ± ( b ; y ) w ith eig en v alues (1.14), where H δ ( b ; z ) ≡ w ( b ; z ) 1 / 2 A δ ( b ; z ) w ( b ; z ) − 1 / 2 = X τ =+ , −  s δ ( z − τ ib ) s δ ( z )  1 / 2 T z τ ia − δ  s δ ( z + τ ib ) s δ ( z )  1 / 2 . (2.69) F rom the G -A∆Es (A.2) w e deduce w ( b ; x ) = 4 s + ( x ) s − ( x ) w r ( b ; x ) , w r ( b ; x ) ≡ G ( ± x − ia + ib ) . (2.70) F or b ∈ (0 , 2 a ) the reduced weigh t fu nction w r ( b ; x ) is p ositiv e f or all real x , and since it is also ev en, its p ositiv e s q u are r o ot f or x > 0 has a real-analytic extension to an ev en p ositiv e function on all of R . By co n trast, it is clear from (2.70) that w ( b ; x ) 1 / 2 , x > 0, extends to an o dd real-analytic function on R . As a consequence, one can also view the transform asso ciated with F( b ; x, y ), b ∈ (0 , 2 a ), as a unitary transform from th e o dd subsp ace of L 2 ( R , dy ) on to the o dd subsp ace of L 2 ( R , dx ). This is the viewp oint take n in [12 ], wh er e we studied this tr an s form (among other ones) for the sp ecial b -v alues N a + with N ∈ N ∗ . As sh own there, for b > 2 a unitarit y and self-adjoin tness generically break down in a wa y that can b e understo o d in great detail. T o b e sure, th e pr ecise connection b et w een the ab o v e functions F and R r and the fu nctions F r and E r from [12] is not clear at face v alue. But the latter are derive d fr om the functions M (( N + 1) a + ; x, y ) of [11], as sp ecif ied b elo w equation (1.42) in [12], so th is connection is enco ded in the identit ies (2.27) for the sp ecial cases ( m, n ) = ( N + 1 , 0), N ∈ N . 2.3 The iden tities (2.27) and t heir consequences W e pro ceed to pro v e the general identitie s (2.27). Our reasoning inv olves in particular a com- parison of the b eha vior for x → ∞ of the fu nctions on the l.h .s. and r.h.s. F or R r ( b ; x, y ) this asymptotics easily follo ws up on com bining (2.39 ), (2.54) and (2.61): R r ( b ; x, y ) ∼ exp( − αbx/ 2) X τ =+ , − c ( b ; τ y ) exp ( τ iαxy / 2) , (2.71) b ∈ R , y > 0 , x → ∞ . Next, w e consider the fun ctions M ( b mn ; ± x, y ). T o b egin with, they are eigenfunctions of the four A∆ O s (1.13) (where b = b mn ) with eigen v alues (1.14), cf. Theorem I I.3 in [11]. Their ‘elementary’ form (2.29) follo ws from equations (3.65)– (3.68) in [11]. The f unction K N + ,N − ( a + , a − ; x, y ) o ccurring in th ese form ulas is sp ecif ied in equation (3.2), with S N δ giv en by equation (2.21). In tur n, the co ef f icien ts in equation (2.21) are def in ed via equations (2.2) –(2.5) in [11]. (S ee also Su bsection 4.1 f or more information on these sp ecial cases.) It is straigh tfor- w ard to obtain th e asymptotics for Re ( x ) → ∞ f r om these exp licit form ulas. Sp ecif ically , this yields M ( b mn ; ± x, y ) = exp( − αb mn x/ 2) c ( b mn ; ± y ) e ± iαxy / 2  1 + O  e − ρ Re ( x )  , (2.72) y > 0 , Re ( x ) → ∞ . The deca y rate ρ is the minimum of the t w o num b ers 2 π /a ± , and the imp lied constan t can b e c hosen uniform for Im ( x ) v aryin g o ver R . Comparing (2.71) and (2.72), it follo ws th at the fu n ctions on the l.h.s. and r.h.s. of (2.27) ha v e the same asymp totics for x → ∞ . It therefore su f f ices to pr o v e that for f ixed a + , a − , y > 0 14 S. Ruijsenaars and m, n ∈ Z , they m ust b e prop ortional as functions of x . Moreo v er, we ma y as w ell assume a + /a − is irrational, sin ce equalit y for this case en tail s equalit y for all a + , a − > 0. (Indeed, the functions M ( b mn ; ± x, y ) are manifestly r eal-analytic in a + and a − for a + , a − > 0, and this real-analytic it y p rop erty is also v alid for R r ( b ; x, y ), cf. I.) The key consequ en ce of the irrationalit y assum p tion is that the v ector space of meromorphic join t solutions f ( x ) to the A∆Es A ± ( ma + + na − ; x ) f ( x ) = 2 c ± ( y ) f ( x ) , m, n ∈ Z , y > 0 , a + /a − / ∈ Q , (2 .73) is t w o-dimensional. T o explain why this is so, w e f irs t n ote that the functions M ( b mn ; ± x, y ) are indep end en t s olutions to (2.73 ), their indep endence already b eing clea r fr om their general form (2.29). Moreo v er, it follo ws from their u niform asymptotics (2.72) that there exists a p o- sitiv e n um b er Λ, dep ending on the f ixed v ariables a + , a − , m , n and y , but not on x , su c h th at in the h alf p lane Re ( x ) > Λ b oth functions are zero-free, and s atisfy lim Im ( x ) →∞ M ( b mn ; x, y ) / M ( b mn ; − x, y ) = 0 , Re ( x ) ∈ ( c − , c + ) ⊂ [Λ , ∞ ) . (2.74) W e are no w in the p osition to in v ok e a result from Section 1 in [23], to the ef fect that the ab o v e su f f ices for an y join t meromorph ic solution f ( x ) of (2.73) to b e a linear combinatio n of the t wo fun ctions M ( b mn ; ± x, y ). Sin ce R r ( b mn ; x, y ) is an ev en meromorph ic joint solution, the f unctions on the l.h.s. and r.h.s. of (2.27) are prop ortional, so their equalit y no w follo ws. In particular, for the free case m = n = 0 we reco v er the id entit y (2.58) from (2.27 ) –(2.28). Moreo v er, taking y = ib mn in (2.2 7) , we can inv oke (2.25) to deduce the corollary M ( b mn ; x, ib mn ) + M ( b mn ; − x, ib mn ) = G ( i ( m − 1 / 2) a + + i ( n − 1 / 2) a − ) G ( i (2 m − 1 / 2) a + + i (2 n − 1 / 2) a − ) . (2.75) Using the G -A∆Es (A.2), the r.h.s. can b e rewritten in terms of sine-functions. F or the s p ecial case m = N + 1, n = 0, the resulting id en tit y amounts to equation (2.78) in [11], cf. also Subsection 4.1. W e would lik e to add in passing th at it is ve ry plausible that (2.74) is not necessary f or t w o- dimensionalit y . In d eed, denoting b y P c the f ield of meromorphic functions with p erio d c ∈ C ∗ , an y third indep endent join t meromorphic solution wo uld ha v e to b e b oth of the form f ( x ) = p 1 ( x ) M ( b mn ; x, y ) + p 2 ( x ) M ( b mn ; − x, y ) , p 1 , p 2 ∈ P ia + , (2.76) and of the form f ( x ) = q 1 ( x ) M ( b mn ; x, y ) + q 2 ( x ) M ( b mn ; − x, y ) , q 1 , q 2 ∈ P ia − . (2.77) Since th e intersect ion of the f ields P ia + and P ia − reduces to the constan ts when a + /a − is irrational, we exp ect (but are unable to p ro v e) th at this sim ultaneous represen tatio n sh ould lead to a con tradiction without app ealing to (2.74). No w that we ha v e pro v ed (2.27), it follo ws th at the fun ction R r ( a + , a − , b ; x, y ), w hic h is real-analytic on the parameter set Π ≡  ( a + , a − , b ) ∈ (0 , ∞ ) 2 × R  , (2.78) is the con tin uous in terp olation of th e functions on the r.h.s. of (2.27), whic h are only def ined for the d en se su bset of ‘elemen tary’ parameters Π el ≡ { ( a + , a − , b ) ∈ Π | b = m a + + na − , m, n ∈ Z } . (2.79) A Relativi stic Conical F unction and its Whittak er Limits 15 The natur al question wh ether another linear com b ination of M ( b mn ; ± x, y ) th at is in dep end en t from the ev en one admits a conti n uous interp olation as we ll remains op en. In this connection w e should p oint out that our reasoning at the en d of Section 3 of [24] r enders this extremely un likely , but is not conclusiv e. Ind eed, w e cannot rule ou t that the s equence of functions Q − giv en b y equation (3.15) in [24], with N + ∈ N , giv es rise to an in f init y of distinct limits L − , corresp ondin g to distinct subsequences. (This o v ersigh t is of no consequen ce for the later sections in [24].) F or the same r eason, the analogous assertion ab out the R -fun ction, made at the end of [10], has not b een completely pro v ed. Before turn ing to parameter shifts, w e derive a non-obvio us r ealit y feat ure of R r from the relations (2.27), namely , R r ( a + , a − , b ; x, y ) = R r ( a + , a − , b ; x, y ) , ∀ ( a + , a − , b, x, y ) ∈ Π × R 2 . (2.80) The p oin t is that from the explicit form ulas for the functions M it is apparent that w e hav e M ( b mn ; x, y ) = M ( b mn ; − x, y ) , x, y ∈ R , (2.81 ) cf. equation (3.73) in [11]. Ther efore, (2.80) is clear from (2.27) and interp olation. As a corollary , this yields realit y of R and F for real parameters and v ariables, cf. (1.6) and (2.68). (Alter- nativ ely , th is realit y prop erty of the R -function follo ws from that of the R -function pr o v ed in Lemma 2.1 of I I I.) 2.4 P arameter shifts W e con tin ue to su mmarize results concerning parameter shifts from [8], inasmuc h as th ey apply to the p r esen t A 1 con text. In S ection 1 of [8] w e introdu ced th e up-shifts S ( u ) δ ( x ) ≡ − i 2 s δ ( x )  T x ia − δ − T x − ia − δ  , (2.82) satisfying S ( u ) δ ( x ) A δ ′ ( b ; x ) = A δ ′ ( b + a δ ; x ) S ( u ) δ ( x ) , (2.83) and the down-shifts S ( d ) δ ( b ; x ) ≡ 2 i s δ ( x )  s δ ( x − ib ) s δ ( x + ia − δ − ib ) T x ia − δ − ( i → − i )  , (2.84) satisfying S ( d ) δ ( b ; x ) A δ ′ ( b ; x ) = A δ ′ ( b − a δ ; x ) S ( d ) δ ( b ; x ) , (2.85) where δ, δ ′ = + , − . Clearly , the up-shifts S ( u ) + ( x ) and S ( u ) − ( x ) comm ute, and the d o wn-shifts S ( d ) + ( b 1 ; x ) and S ( d ) − ( b 2 ; x ) comm ute as well. T he shifts are also related by S ( u ) δ ( x ) S ( d ) δ ( b ; x ) = A δ ( b ; x ) 2 − 4 cos 2 ( π ( b − a − δ ) /a δ ) , (2.86) S ( d ) δ ( b + a δ ; x ) S ( u ) δ ( x ) = A δ ( b ; x ) 2 − 4 cos 2 ( π b/a δ ) , (2.87) where δ = + , − . It is a matter of straigh tforw ard calculatio ns to v erify the form ulas (2.83) and (2.8 5)–(2.87 ). Starting from the join t eigenfunctions exp( ± iαxy / 2) of A ± (0; x ) with eigen v alues 2 c ± ( y ), one can no w obtain j oin t eigenfunctions with the same eigen v alues for A ± ( b mn ; x ) by acting with 16 S. Ruijsenaars the shifts on the plane wa ves. By constr u ction, these join t eigenfun ctions are of the elemen- tary form (2.29). Cho osin g a + /a − irrational, it follo ws from tw o- dimensionalit y of the join t eigenspace that these eigenfunctions are (generally y -dep endent) multiples of M ( b mn ; ± x ; y ). A more telling action of the shifts is en co ded in S ( u ) δ ( x ) R r ( b ; x, y ) = 4 s δ ( y + ib ) s δ ( y − ib ) R r ( b + a δ ; x, y ) , (2.88) S ( d ) δ ( b ; x ) R r ( b ; x, y ) = R r ( b − a δ ; x, y ) . (2.89) Indeed, th ese relations hold for arbitrary b . F or b = b mn , it then follo ws by using (2.27) that they also hold for the summand s M ( b mn ; ± x, y ). (This is b ecause their plane wa ve factors are indep en d en t, cf. (2.29).) The equations (2.89) and (2.88) f ollo w from a suitable sp ecializat ion of equations (3.11) and (3.13) in [8]. But in the presen t A 1 case we can also der ive them quite easily by using the elementa ry join t eige nfunctions M ( b mn ; x, y ) with a + /a − / ∈ Q . Indeed, once w e ha v e sho wn that (2.89), (2.88) hold for y > 0, b = b mn , m, n ∈ Z , and with R r replaced b y M , it is easy to dedu ce (2.89), (2.8 8) from (2.27) and interp olation. (Note that the four shifts commute with parit y .) Th eir v alidit y for these sp ecial cases can b e readily v erif ied: O ne need only show that the functions on the l.h.s. and r.h.s. ha v e the same x → ∞ asymptotics, and using (2.72) this causes little d if f icult y . Next, w e obtain the counterparts of the A δ ( b ; x )- and R r -shifts for the A∆Os A δ ( b ; x ) and their joint eige nfun ction E( b ; x, y ). (F or the B C 1 setting we did this in Section 8 of [9]; as in previous cases, it is in fact simpler and more illuminating to obtain the r elev an t form ulas by direct means, instead of b y s p ecializatio n.) T hey are give n b y S ( u ) δ ( b ; x ) ≡ 1 c ( b + a δ ; x ) S ( u ) δ ( x ) c ( b ; x ) , (2.90) S ( d ) δ ( b ; x ) ≡ 1 c ( b − a δ ; x ) S ( d ) δ ( b ; x ) c ( b ; x ) . (2.91) A moment’ s though t sho ws th at this en tails the v alidit y of (2.85)–(2.87) with S , A rep laced b y S , A . Also, using the def inition (2.41) of the c -fu nction and the G -A∆Es (A.2), we obtain the explicit formulas S ( u ) δ ( b ; x ) = T x ia − δ − s δ ( x − ib ) s δ ( x − ib + ia − δ ) s δ ( x ) s δ ( x + ia − δ ) T x − ia − δ , (2.92) S ( d ) δ ( b ; x ) = T x ia − δ − s δ ( x + ib ) s δ ( x + ib − ia − δ ) s δ ( x ) s δ ( x + ia − δ ) T x − ia − δ . (2.93) Notice th at they imp ly S ( u ) δ (2 a − b ; x ) = S ( d ) δ ( b ; x ) . (2.94) Finally , a straigh tforw ard calculation yields the coun terparts of (2.88) and (2.89): S ( u ) δ ( b ; x )E( b ; x, y ) = 2 e δ ( − iπ b ) s δ ( y + ib )E( b + a δ ; x, y ) , (2.95) S ( d ) δ ( b ; x )E( b ; x, y ) = 2 e δ ( iπ ( b − a − δ )) s δ ( y − ib + ia − δ )E( b − a δ ; x, y ) . (2.96) T o conclude th is secti on, w e p oin t out th at the eigh t sh ifts acting on x ha v e d u als acting on y giv en b y the form ulas (2.82), (2.84) and (2.90)–(2.93) w ith x → y . By self-du ality , their resp ectiv e actions on R r ( b ; x, y ) and E( b ; x, y ) follo w from the ab o v e b y interc h an ging x and y . A Relativi stic Conical F unction and its Whittak er Limits 17 3 Fiv e minimal repr esen tations of the R -function W e b egin this section b y fo cusing on the k ernel function K ( b ; x, v ) := G (( ± x ± v − ib ) / 2) ≡ Y δ 1 ,δ 2 =+ , − G (( δ 1 x + δ 2 v − ib ) / 2) . (3.1) W e hav e established that this function satisf ies three indep endent kernel iden tities. W e exp ect that these might b e useful in other con texts than the pr esent one. Indeed, here we only need th e sp ecial case (3.14) of the f irs t of the identitie s. W e colle ct the three iden tities in the follo wing prop osition. Prop osition 3.1. L etting b, d ∈ C , we have s δ ( x − ib + id ) s δ ( x ) K ( b ; x − ia − δ , v ) + ( i → − i ) = s δ ( v − id ) s δ ( v ) K ( b ; x, v − ia − δ ) + ( i → − i ) , (3.2) s δ ( x − ib ) s δ ( x ) K ( b ; x − 2 ia − δ , v ) + ( i → − i ) = s δ ( v − ib ) s δ ( v ) K ( b ; x, v − 2 ia − δ ) + ( i → − i ) , (3.3) s δ (( x − ib ) / 2) s δ ( x/ 2) K ( b ; x − ia − δ , v ) + ( i → − i ) = s δ (( v − ib ) / 2) s δ ( v / 2) K ( b ; x, v − ia − δ ) + ( i → − i ) . (3.4) Pro of . T o p ro v e (3.2 ), w e divide l.h.s. and r.h .s. by K ( b ; x − ia − δ , v ) and use the A∆Es (A.2) to write the result as s δ ( x − ib + id ) s δ ( x ) + s δ ( x + ib − id ) s δ ( x ) c δ (( x + v − ib ) / 2) c δ (( x + v + ib ) / 2) c δ (( x − v − ib ) / 2) c δ (( x − v + ib ) / 2) = s δ ( v − id ) s δ ( v ) c δ (( x − v − ib ) / 2) c δ (( x − v + ib ) / 2) + s δ ( v + id ) s δ ( v ) c δ (( x + v − ib ) / 2) c δ (( x + v + ib ) / 2) . (3.5) Both sides are 2 ia δ -p erio dic functions of x with equal limits e δ ( ± ( − ib + id )) + e δ ( ± ( − ib − id )) , Re x → ±∞ . (3.6) The residu es at the (generical ly simple) p oles x = 0, x = ia δ in the p erio d strip clearly cancel. By Liouville’s theorem, it remains to c hec k that the residu es at the p oles x = ± v − ib/ 2 ± ia δ cancel as wel l, and this is a routine cal culation. Next, w e d ivide (3. 3) b y K ( b ; x, v ) an d use (A.2) to obtain s δ ( x − ib ) s δ ( x ) c δ (( x − ia − δ / 2 ± v + ib ) / 2) c δ (( x − ia − δ / 2 ± v − ib ) / 2) + ( x → − x ) = s δ ( v − ib ) s δ ( v ) c δ (( v − ia − δ / 2 ± x + ib ) / 2) c δ (( v − ia − δ / 2 ± x − ib ) / 2) + ( v → − v ) . (3. 7) Both sides are 2 ia δ -p erio dic functions of x with equal limits e δ ( ± ib ) + e δ ( ∓ ib ) = s δ ( v − ib ) s δ ( v ) + s δ ( v + ib ) s δ ( v ) , Re x → ±∞ . (3.8) 18 S. Ruijsenaars The residues at x = 0 and x = ia δ manifestly cancel. It is a straigh tforw ard calculation to v erify that the resid u es at the remaining p oles x = ± ia δ ± ia − δ / 2 ± v + ib cancel, to o. Hence (3.3) follo ws. Finally , to prov e (3.4), we divid e b oth sides b y K ( b ; x − ia − δ , v ) and use (A.2) to get as the coun terpart of (3.5): s δ (( x − ib ) / 2) s δ ( x/ 2) + s δ (( x + ib ) / 2) s δ ( x/ 2) c δ (( x + v − ib ) / 2) c δ (( x + v + ib ) / 2) c δ (( x − v − ib ) / 2) c δ (( x − v + ib ) / 2) = s δ (( v − ib ) / 2) s δ ( v / 2) c δ (( x − v − ib ) / 2) c δ (( x − v + ib ) / 2) + s δ (( v + ib ) / 2) s δ ( v / 2) c δ (( x + v − ib ) / 2) c δ (( x + v + ib ) / 2) . (3.9) Both sides are 2 ia δ -p erio dic functions of x with equal limits e δ ( ± ( − ib/ 2)) + e δ ( ± ( − 3 ib/ 2)) , Re x → ±∞ . (3 .10) As b efore, residue cancellation at x = 0 and x = ia δ is immediate, whereas the v erif ication that the residues at th e remaining p oles x = ± ia δ ± v − ib/ 2 cancel as w ell in v olv es a bit m ore w ork.  F rom (1.7) w e see that the id en tit y (3.2) can b e rewritten as A δ ( b − d ; x ) K ( b ; x, v ) = A δ ( d ; v ) K ( b ; x, v ) , δ = + , − . (3.11) A t f irst sigh t, one migh t think that the identiti es (3.3 ) and (3.4) can also b e rewr itten by u s ing a rescaled v ersion of the t w o commuting A 1 dif ference op erators A ± ( b ; z ). The t w o dif ference op erators s δ (( z − ib ) / 2) s δ ( z / 2) T z ia − δ + s δ (( z + ib ) / 2) s δ ( z / 2) T z − ia − δ , δ = + , − , (3.12) featuring in (3.4), do not ev en comm ute, h o w ev er. Thus no suc h rescaling is p ossible for (3.4). The dif ference op erators s δ ( z − ib ) s δ ( z ) T z 2 ia − δ + s δ ( z + ib ) s δ ( z ) T z − 2 ia − δ , δ = + , − , (3.13) corresp ondin g to (3.3) do commute. Ev en so, one can only rescale one of the op erators such that it tak es the A 1 form (1.7), but not b oth at once. F or the pu rp ose of studying the A 1 op erators, then, we can only m ake use of (3.2). More sp ecif ically , our starting p oin t is the sp ecial case d = 0: A δ ( b ; x ) K ( b ; x, v ) =  T v ia − δ + T v − ia − δ  K ( b ; x, v ) . (3.14) In order to exploit this iden tit y , w e introdu ce the F ourier transform B ( b ; x, y ) ≡ 1 2 Z R dv K ( b ; x, v ) exp ( iαv y / 2) , b ∈ (0 , 2 a ) , x, y ∈ R . (3.15) The in teg ral is w ell d ef ined, since the b -restriction ensures that the v -p oles of the in tegrand at ± v = x + 2 ia − ib + z k l , ± v = − x + 2 ia − ib + z k l , k , l ∈ N , (3.16) (cf. (A.17)–(A.16)), sta y a w a y from the real axis, and since the G -asymptotics (A.13) ent ails an exp onenti al deca y K ( b ; x, v ) ∼ exp( ∓ αbv / 2) , b > 0 , x ∈ C , Re v → ±∞ . (3.17) A Relativi stic Conical F unction and its Whittak er Limits 19 These features also imply that B extends from the real x -axis to a f u nction that is holomorphic in the strip Im x ∈ ( − 2 a + b, 2 a − b ). Next, w e temp orarily assume b ∈ (0 , a s / 2) , a s ≡ min( a + , a − ) . (3.18) Then the action of th e shifts in the A∆ O s A ± ( b ; x ) giv en by (1.7) is w ell d ef ined on B ( b ; x, y ), pro vided we restrict x to a strip | Im x | < a s / 2. Moreo v er, we ma y tak e the shifts un der the in tegral sign in (3.15) and u se th e ke rnel iden tit y (3.14) to obtain A δ ( b ; x ) B ( b ; x, y ) = 1 2 Z R dv X τ =+ , − K ( b ; x, v + τ ia − δ ) exp( iπ v y /a + a − ) , (3.19) | Im x | < a s / 2 . Up on shifting co n tours R → R ± ia − δ , n o p oles are met, and so w e obtain 1 2 X τ =+ , − e δ ( τ y ) Z R − iτ a − δ dv K ( b ; x, v ) exp ( iπ v y /a + a − ) . (3.20) The integrands of b oth terms are no w equ al, and the conto urs can b e shifted b ac k to R without c hanging the v alue of the in tegral s. Hence w e deduce the eigen v alue equations A ± ( b ; x ) B ( b ; x, y ) = 2 c ± ( y ) B ( b ; x, y ) , | Im x | < a s / 2 . (3.21) Rev erting to our previous assumption b ∈ (0 , 2 a ), w e pro ceed to obtain t w o dif ferent repre- sen tations of B ( b ; x, y ). T o this end we us e the Planc herel relation Z R dpf ( p ) g ( p ) = α 2 π Z R dq ˆ f ( q ) ˆ g ( − q ) , f , g ∈ L 2 ( R ) ∩ L 1 ( R ) , (3.22) with the F our ier transform d ef ined by ˆ h ( q ) = Z R dp exp( iαpq ) h ( p ) , h = f , g . (3.23) Rewriting B as B ( b ; x, y ) = Z R dp G ( p + ( x − ib ) / 2) G ( p − ( x + ib ) / 2) exp ( iαpy ) G ( p − ( x − ib ) / 2) G ( p + ( x + ib ) / 2) , (3.24) b ∈ (0 , 2 a ) , x, y ∈ R , w e no w def ine f 1 ( p ) ≡ G ( p + ( x − ib ) / 2) G ( p − ( x − ib ) / 2) , f 2 ( p ) ≡ G ( p + ( x − ib ) / 2) G ( p + ( x + ib ) / 2) , (3.25) g 1 ( p ) ≡ G ( p − ( x + ib ) / 2) exp ( iαpy ) G ( p + ( x + ib ) / 2) , g 2 ( p ) ≡ G ( p − ( x + ib ) / 2) exp ( iαpy ) G ( p − ( x − ib ) / 2) . (3.26) W e can calculate the F ourier tr an s forms of these four functions b y using Prop osition C.1. Doing so, w e use the Plancherel relation (3.22) and th en replace q by z + y / 2 to obtain the t w o represent ations announced abov e: B ( b ; x, y ) = G ( ± x + ia − ib ) Z R dz G ( z ± ( x − y ) / 2 − ia + ib/ 2) G ( z ± ( x + y ) / 2 + ia − ib/ 2) , (3.27) 20 S. Ruijsenaars B ( b ; x, y ) = G ( ia − ib ) 2 Z R dz G ( ± z ± y / 2 − ia + ib/ 2) exp( iαz x ) . (3.28) Next, w e compare (3.28) to (3.24), dedu cing that it can b e rewritten as B ( b ; x, y ) = G ( ia − ib ) 2 B (2 a − b ; y , x ) . (3.29) Also, d ef ining a n ew function C ( b ; x, y ) b y (1.11), w e see that (3.27) amounts to B ( b ; x, y ) = r 2 π α G ( ± x + ia − ib ) C (2 a − b ; x, y ) . (3.30) The function C ( a + , a − , b ; x, y ) is of cen tral imp ortance f or what follo ws. W riting it as C ( a + , a − , b ; x, y ) = r α 2 π Z R dz Y δ =+ , − G ( a + , a − ; z − u δ ) G ( a + , a − ; z − d δ ) , (3.31) where w e hav e introdu ced u ± ≡ ± ( x − y ) / 2 + ib/ 2 , d ± ≡ ± ( x + y ) / 2 − ib/ 2 , (3.32) w e in fer th at its b eha vior un der analytic con tin uation in its 5 v ariables is imm ediate fr om the general analysis in App endix B of I (with N sp ecialized to 2). W e pro ceed to su mmarize the salien t information. T o this end, w e need the function E ( a + , a − ; z ) discussed in App endix A, cf. (A.20). In tro- ducing P ( a + , a − , b ; x, y ) ≡ C ( a + , a − , b ; x, y ) E ( a + , a − ; ± x + ib − ia ) E ( a + , a − ; ± y + ib − ia ) , (3.33 ) the pro du ct fun ction P ( a + , a − , b ; x, y ) extends from (0 , ∞ ) 2 × (0 , a + + a − ) × R 2 to a function that is holomorphic in the domain D ( a + , a − , b ) ≡  ( a + , a − , b, x, y ) ∈ C 5 | Re a + > 0 , Re a − > 0 , Re ( b/a + a − ) > 0  . (3.34) Hence C is meromorphic in D ( a + , a − , b ), with p oles occurr in g solely at the zeros ± x = 2 ia − ib + z k l , ± y = 2 ia − ib + z k l , k, l ∈ N , (3.35) of the E -pro d uct, cf. (A.21); moreo v er, the maximal m ultiplicit y of a p ole at z = z 0 , with z = x, y , is giv en by the zero m ultiplicit y at z = z 0 of the p ertinen t E -factor. The corresp onding meromorph y prop erties of B are no w clear from its relation (3. 30) to C : It con tin ues meromorphically to the d omain D ( a + , a − , 2 a − b ). F rom (3.29) w e then deduce that B ( a + , a − , b ; x, y ) h as a meromorp hic extension to the larger d omain D + (2.26). Using (3.30 ) again, it n o w follo ws that C has a meromorphic extension to D + as w ell. W e pro ceed to obtain f u rther information on the fu nction C . First, w e list f eatures that are immediate from its def inition (1.11) and pr op erties of th e G -function, cf. Ap p end ix A: C ( a + , a − , b ; x, y ) = C ( a − , a + , b ; x, y ) , (mo d ular in v ariance) , (3.36) C ( a + , a − , b ; x, y ) = C ( a + , a − , b ; y , x ) , (self-dualit y) , (3.37) C ( b ; x, y ) = C ( b ; δ x , δ ′ y ) , δ , δ ′ = + , − , (ev enness) , (3.38) C ( a + , a − , b ; x, y ) = C ( λa + , λa − , λb ; λx, λy ) , λ > 0 , (scale in v ariance) , (3.39) C ( a + , a − , b ; x, y ) ∈ R , a + , a − > 0 , b ∈ (0 , 2 a ) , x, y ∈ R , ( real-v aluedness) . (3.40) A Relativi stic Conical F unction and its Whittak er Limits 21 Clearly , the relations (3.36)–(3.39) are we ll def ined and hold true on D + (2.26). Also, the prop erty (3.40) can b e r en dered m an if est by substituting the inte gral representat ion (A.5) in the four G -functions and com bining facto rs to obtain C ( a + , a − , b ; x, y ) = r α 2 π Z R dz cos  Z R dw w sin( xw ) sin( y w ) cosh ( bw ) sin (2 z w ) sinh( a + w ) sinh( a − w )  (3.41) × exp  Z R dw w  cos( xw ) cos ( y w ) sinh ( bw ) cos(2 z w ) sinh( a + w ) sinh ( a − w ) − b a + a − w  , where ( a + , a − , b, x, y ) ∈ (0 , ∞ ) 2 × (0 , 2 a ) × R 2 . Second, com bining (3.37) w ith (3.29) and (3.3 0), we deduce G ( ia − ib ) C ( b ; x, y ) G ( x + ia − ib ) G ( y + ia − ib ) = G ( ib − ia ) C (2 a − b ; x, y ) G ( x − ia + ib ) G ( y − ia + ib ) , ( b -symmetry) . (3.42) Third, w e can use Prop osition C .1 once more to obtain from (1.11) the explicit result C ( b ; x, ib ) = G ( ia − 2 ib ) G ( ib − ia ) 2 , (normaliza tion) , (3.43) where ( a + , a − , b, x ) ∈ (0 , ∞ ) 2 × (0 , a ) × R . Last b ut not least, w e claim that we ha v e the j oint eigen v alue equations A ± ( b ; x ) C ( b ; x, y ) = 2 c ± ( y ) C ( b ; x, y ) , A ± ( b ; y ) C ( b ; x, y ) = 2 c ± ( x ) C ( b ; x, y ) . (3.44) T o pr o v e this claim, w e f irst note that the eigen v alue equ ations (3.21) con tin ue meromorph i- cally to D + . Next, we observe that the G -A∆Es (A.2 ) imp ly G ( ± x + ia − ib ) − 1 A δ ( b ; x ) G ( ± x + ia − ib ) = A δ (2 a − b ; x ) , δ = + , − . (3.45) (Here, we view the l.h.s. as the pro duct of three op erators acting on meromorp hic functions.) Hence w e obtain via (3.3 0) A ± (2 a − b ; x ) C (2 a − b ; x, y ) = 2 c ± ( y ) C (2 a − b ; x, y ) , (3.46) whic h is equiv ale n t to the f irst t w o A∆Es in (3.44). The last tw o are then clear from the self-dualit y relation (3.37) . All of the prop erties of C just der ived also hold true for the fu nction G ( ib − ia ) R r def ined b y (1.6 ) and (1.2), cf. Sectio n 2. By u sing solely the eigenv alue prop erties (3. 44), the ev enness prop erties (3.38), and the normalization (3.43) , w e can no w show that these tw o functions coincide, as announced in the In tro duction, cf. (1.12). Sp ecif ically , app lyin g the uniqueness argumen t in Sub section 2.3 to C in its dep endence on x , we obtain th e equalit y (1.12 ) up to a prop ortionalit y factor p ( a + , a − , b, y ). Rep eating this argument for the y -dep end ence, w e see that the prop ortionalit y factor can only dep end on the parameters a + , a − , b . F rom the normalization relatio n (3.43) it then follo ws that p = 1, thus proving (1. 12). Let us n o w collect the resulting minimal r ep resen tations of the R -fun ction. F rom (1.12 ) and (1.6) we obtain R ( b ; x, y ) = 1 √ a + a − G (2 ib − ia ) G ( ib − ia ) 2 Z R dz G ( z ± ( x − y ) / 2 − ib/ 2) G ( z ± ( x + y ) / 2 + ib/ 2) . (3.4 7) Next, com bining (3.24) and (3.30), w e deduce R ( b ; x, y ) = 1 √ a + a − G (2 ib − ia ) G ( ib − ia ) 2 G ( ± x + ia − ib ) 22 S. Ruijsenaars × Z R dz G ( z ± x/ 2 − ia + ib/ 2) G ( z ± x/ 2 + ia − ib/ 2) exp( iαz y ) , (3.48) and using (3.2 9) we infer R ( b ; x, y ) = 1 √ a + a − G (2 ib − ia ) G ( ± x + ia − ib ) Z R dz G ( z ± y / 2 − ib/ 2) G ( z ± y / 2 + ib/ 2) exp( iαz x ) . (3.49) Finally , using the self-du alit y prop erty of R , w e obtain from (3.48) and (3.49) the rep r esen tations R ( b ; x, y ) = 1 √ a + a − G (2 ib − ia ) G ( ib − ia ) 2 G ( ± y + ia − ib ) × Z R dz G ( z ± y / 2 − ia + ib/ 2) G ( z ± y / 2 + ia − ib/ 2) exp( iαz x ) , (3.50) R ( b ; x, y ) = 1 √ a + a − G (2 ib − ia ) G ( ± y + ia − ib ) × Z R dz G ( z ± x/ 2 − ib/ 2) G ( z ± x/ 2 + ib/ 2) exp( iαz y ) . (3 .51) The f ive representati ons (3.47)–(3.51) are wel l d ef ined and hold true for ( a + , a − , b, x, y ) ∈ (0 , ∞ ) 2 × (0 , 2 a ) × R 2 . (Com bining (3.47) with (3.42) , we get a further representa tion that w e do not consider.) T aking sto c k of the ab o v e develo pment s, w e note that w e migh t hav e started f r om the f irst minimal representati on (3.47) to define the R -fun ction. Then many of its p rop erties follo w quite easil y . On the other hand, it seems n ot feasible to giv e a direct pro of of its cr u cial join t eigenfunction p r op erty . With h indsight, h o w ev er, this can b e shown by f irst obtaining the second repr esen tation (3.48) (sa y) via Pr op osition C.1, and th en using the iden tit y (3.14) to arriv e at (3.21). F rom this the join t eigenfunction prop ert y (3. 44) follo ws as b efore. Another imp ortan t prop ert y of R is its asymptotic b eha vior for x → ∞ . Lik e other features addressed in this section, this can already b e gleaned from Section 2, via the sp ecializat ion of the more general asymptotics of the f unction E ( γ ; v , ˆ v ) obtained in Theorem 1.2 of I I. Ho w ev er, pro vided we restrict b to th e in terv al (0 , 2 a ), it is quite easy to obtain the x → ∞ asymptotics directly from the new represent ations of R in terms of a F ourier transform. T o detail this, let us f irst note that w e n eed only consider the function E( b ; x, y ), whic h w e can no w view as b eing def ined via (2.39)–(2.41). (Indeed, there is n o dif f iculty in obtai- ning the asymptotics of the c -fu n ction; in this connection, compare (2.61), (2.54) and (2.71).) Using (3.49), w e d educe that for b ∈ (0 , 2 a ) the E-function has th e represen tation E( b ; x, y ) = φ ( b ) √ a + a − G ( ib − ia ) c ( b ; y ) G ( x + ia ) G ( x − ia + ib ) Z R dz G ( z ± y / 2 − ib/ 2) G ( z ± y / 2 + ib/ 2) exp( iαz x ) . (3.52) Letting y ∈ (0 , ∞ ), w e can shift the conto ur u p b y a − b/ 2 + ǫ , where ǫ > 0 is small enough so that only th e simp le p oles at z = ± y / 2 − ib/ 2 + ia, (3.53) are encoun tered. The residu es at these p oles easil y follo w from (A.19 ), yielding a contribution M ( b ; x )(exp ( iαxy / 2) + c ( b ; − y ) exp( − iαxy / 2) /c ( b ; y )) , (3.54) with the multiplier giv en by M ( b ; x ) ≡ φ ( b ) G ( x + ia ) G ( − x + ia − ib ) exp ( − αx ( a − b/ 2)) . (3.55) A Relativi stic Conical F unction and its Whittak er Limits 23 No w f rom (A.13) we see that M ( b ; x ) con v erges to 1 for x → ∞ . T o reco v er the asymp- totics (2.54 ), therefore, it is enough to sho w that the r .h.s. of (3.52) with z replaced by z + ia − ib/ 2 + iǫ v anishes for x → ∞ . T o pro v e th is, w e wr ite the shifted con tour in teg ral as 1 √ a + a − G ( ib − ia ) c ( b ; y ) M ( b ; x ) exp ( − ǫαx ) Z R dz G ( z + iǫ ± y / 2 − ib + ia ) G ( z + iǫ ± y / 2 + ia ) exp( iαz x ) . (3.56) No w one need only u se (A.13) to verify that the in tegrand is b ounded by a m ultiple of exp( − αb | z | / 2), wh ic h imp lies the fun ction (3.56) do es con v erge to 0 for x → ∞ . W e stress that this short argument only yields (2.54) und er the restriction b ∈ (0 , 2 a ). In particular, by con trast to th e previous conto ur inte gral representat ion used in I I, one must cop e with an inevitable con tour pinc hing w hen one tries to use (3.52) to go b ey ond this b -int erv al. Another iss ue is that stron ger asymptotic estimates than just obtained are n ecessary to reco v er the Hilb ert space transform f eatures for the E-function sketc hed in Su bsection 2.3, cf. (2.5 5)–(2.60 ). It is b ey ond our scope to study this fu rther, but w e would like to rep eat that the b -in terv al [0 , 2 a ] cannot b e enlarged without losing th e critical un itarit y and self-adjointness prop erties [12]. A t face v alue, the n ew rep r esen tations (3.47)–(3.51) seem to hold pr omise for a direct pro of of the shift prop erties of R r , cf. (2.88)–(2.8 9 ). Even so, we were u nable to pu s h this through. T o d ate, therefore, th e only reasoning yielding the prop erties for general b is to f irst derive th em for sp ecial b -v alues, as sketc hed in S ubsection 2.4. W e no w turn to a study of the R r -function for these sp ecial v alues. 4 Sp ecializations and nonrelativistic limit 4.1 Elemen tary sp ecial cases As already mentio ned, the functions R N ( a + , a − ; x, y ) ≡ R r ( a + , a − , ( N + 1) a + ; x, y ) , N ∈ N , (4.1) ha v e b een extensiv el y studied b efore. Th ey w ere f irst ob tained more than tw ent y y ears ago [25], and then reconsidered fr om an algebraic and fu nction-theoretic viewp oint in [11] and fr om a representa tion-theoretic viewp oint in a pap er by v an Diejen and Kirillo v [26]. Th e corre- sp ond ing Hilb ert space transforms were s tudied in great detail in [12]. Our f irst goal in this section is to demonstrate h o w the elemen tary c haracter of these f u nctions can b e d irectly understo o d from the F ourier transform representat ions (3.48)–(3.51). Indeed, th us far the relation enco ded in (2.27) has on ly b een sho wn b y app ealing to a un iqu eness argumen t. Th e crux is that for the c hoices b = ( N + 1) a ± one can use the G -A∆Es (A.2) to obtain in tegral s that can b e explicit ly ev aluated by a residue calcula tion. Sp ecif ically , let us start fr om (3.49) to obtain f ir st R N ( x, y ) = 4 − N − 1 √ a + a − G ( i ( N + 1) a + − ia ) G ( ± x + ia − i ( N + 1) a + ) × Z R dz e iαz x / N Y j =0 c − ( z ± y / 2 − i ( N − 2 j ) a + / 2) . (4.2) No w w e r ecall that (3.49) is v ali d for b ∈ (0 , 2 a ), whic h implies w e h a v e N a + < a − in (4.2). T aking y > 0 from n o w on, it follo ws that the in tegrand has 2 N + 2 simple p oles in the strip 24 S. Ruijsenaars Im z ∈ (0 , a − ). The pro du ct yields a function that is ia − -p erio dic in z . Th us, denoting the in tegral b y I N , we h av e I N − exp( − 2 π x/a + ) I N = 2 π i X δ =+ , − N X j =0 Res  z = 1 2  δ y + ia − + i ( N − 2 j ) a +   . (4.3) The residues are easily calculat ed, and hence w e obtain I N = ( − i ) N +1 a − s + ( x )    e iαyx / 2 s − ( y ) N X j =0 e − ( − ( N − 2 j ) x ) Q k 6 = j s − ( y + i ( k − j ) a + ) sin( π ( j − k ) a + /a − ) +( y → − y )    . (4.4) Also, the prefactor can b e calculated by using (A.2) once more, com b ined with (A.12). Intro- ducing P N ( z ) ≡ N Y j = − N 2 s − ( z + ij a + ) , z = x, y , (4.5) w e get 2 − 2 N − 1 a − s + ( x ) P N ( x ) N Y j =1 2 sin ( π j a + /a − ) . (4.6) F or N = 0 the pr o duct of (4.4) and (4.6) yields R 0 ( x, y ) = sin( π xy /a + a − ) 2 s − ( x ) s − ( y ) . (4.7) More generall y , the pro d uct can b e written as R N ( x, y ) = ( − i ) N +1 ( K N ( x, y ) − K N ( x, − y )) /P N ( x ) P N ( y ) , (4.8) where w e hav e set K N ( x, y ) ≡ exp( ixy /a + a − ) N Y l =1 2 sin( π l a + /a − ) × N X j =0 e − ((2 j − N ) x ) N Q k = j +1 s − ( y − ik a + ) N Q k = N − j +1 s − ( y + ik a + ) Q k 6 = j sin( π ( j − k ) a + /a − ) . (4.9) In tro ducing the p hase f actor q ≡ exp( iπ a + /a − ) , (4.10) it is not hard to s ee from (4.9) that K N is of the form K N ( x, y ) = exp( ixy /a + a − ) e − ( N x + N y ) S N ( q ; e − ( − 2 x ) , e − ( − 2 y )) , (4.11) S N ( q ; r, t ) = N X k ,l =0 c ( N ) k l ( q ) r k t l , (4.12) with c ( N ) k l ( q ) a r ational fun ction of q . A Relativi stic Conical F unction and its Whittak er Limits 25 Th us far, our conclusions ab out R N and K N w ere only b ased on an explicit ev al uation of the represent ation (3.49). Ho w ev er, a lot more inform ation follo ws up on using th e features of R r ( b ; x, y ). In particular, th e fun ction K N ( x, y ) /P N ( x ) P N ( y ) is a joint eigenfunction of the four A∆Os (1.13) (where b = ( N + 1) a + ) with eigen v alues (1.14), since this holds true for R N ( x, y ). Also, th e self-dualit y and ev enness of R N imply K N ( x, y ) = K N ( y , x ) = K N ( − x, − y ) , (4.13) and this enta ils that the coef f icients in (4.12) ha v e the sym m etry p rop erties c ( N ) k l = c ( N ) lk = c ( N ) N − k ,N − l , k, l = 0 , . . . , N . (4.14) Of course, this ca n b e easily c hec k ed for sm all N , but f or arbitrary int egers (4.14) is not at all ob vious from (4.9). One m ore feature of the co ef f icien ts is that they are Lauren t p olynomials in q with in teger co ef f icien ts. Like the symmetries (4.14), it is not a routine matter to sho w th is f r om (4.9). The crux is, how ever, that the ab o v e fu nctions K N coincide with those of [11], by virtue of the u niqueness argument used in Subsection 2.3. The co ef f icien ts were studied in detail in Section I I of [11], and there the in terested reader can f in d explicit form ulas for the co ef f icients as Lauren t p olynomials in q . See also the pap er b y v an Diejen and Kirillo v [26], w here yet dif ferent repr esen tations of the functions K N ( x, y ) w ere obtained. With a little more ef fort, the element ary c haracter of R r for the more general b -v alues b + − ≡ ( N + 1) a + − M a − , N , M ∈ N , (4.15) can also b e u n dersto o d from (3.49 ). Indeed, from (A.2) it follo ws by a straigh tforw ard calculation that w e hav e an ident it y G ( v − ib + − / 2) G ( v + ib + − / 2) = M Q k =1 2 c +  v + i 2  ( N + 1) a + + ( M + 1 − 2 k ) a −  N Q j =0 2 c −  v + i 2  M a − + ( N − 2 j ) a +  . (4.16) Using this iden tit y sev eral times (toget her with (A.1 2) for the factor G ( ib + − − ia )), w e deduce from (3.4 9) and (1.6) the representat ion R r ( b + − ; x, y ) = 1 a − N Q j =1 2 sin ( j π a + /a − ) M Q k =1 2 sin( k π a − /a + ) M Q k = − M 2 s + ( x − ik a − ) N Q j = − N 2 s − ( x − ij a + ) (4.17) × Z R dz e iαz x M Q k =1 4 c +  z + 1 2  ± y + i ( N + 1) a + + i ( M + 1 − 2 k ) a −  N Q j =0 4 c −  z + 1 2  ± y + iM a − + i ( N − 2 j ) a +  . The denominator of the in tegrand h as n o zero for z ∈ R unless M is o dd and N is ev en. The zeros of the corresp ond ing factor s − ( z ± y / 2) are th en matc hed by the zeros of the factor s + ( z ± y / 2) of the n umerator. After a suitable cont our shift, w e can expand the numerato r pro du ct into exp onenti als, yielding a sum of con vergen t in tegrals (reca ll w e require b + − ∈ (0 , 2 a )). When M is ev en or N is o dd , w e can do the s ame without a con to ur shift. Eac h of the in tegral s is then 26 S. Ruijsenaars basically of the same form as p r eviously ev aluated for the case M = 0. (More in detail, the same 2 N + 2 p oles arise in the p erio d s trip for the c − -pro du ct.) F rom these observ ations the general structure an ticipated in (2.27) and (2.29) readily follo ws, pro vided n > 0 and m ≤ 0, or m > 0 and n ≤ 0. In Section I I I of [11] we studied the functions (4.17) in considerable detail, but it is b eyo nd our scop e to d er ive the explicit form used there directly from their represen tatio n (4.17 ). W e do add that it seems plausible that the factorization exhibited in equations (3.3)–(3.4) of [11] can b e und ersto o d b y a more ref ined analysis of the ab ov e sum of conto ur integral s. In an y case, w e rep eat that equalit y of the p ertinent functions follo w s from the u n iqueness argum en t explained in Sub section 2.3. 4.2 The nonrelativistic limit W e b egin this su b section with a remark addressed to physicist readers, who ma y care ab out dimension issu es. In our pap er [11], whic h we had occasion to cite several times in the previous subsection, the v ariable y of the present pap er was denoted by p . This is a w idely used notation for the sp ectral v ariable in n onrelativistic quantum mec hanics, where p is view ed as a m omen tum. In our relativistic setting, how ever, it is far more n atural to view the scale parameters a + and a − as ha v in g dimension [position], and then the ‘geo metric’ and ‘sp ectral’ v ariables x and y b oth ha v e d imension [p osition] as we ll. (T o b e more sp ecif ic, one of the scale parameters can b e view ed as an in teractio n range, and the other one as the C ompton wa ve length ~ /mc of the relativistic particles u nder consideration.) This go es to explain our c hange from p to y . Of course, from a mathematical viewp oin t suc h notatio nal issues and the n otion of dimension ma y b e ignored . When taking th e nonrelativistic limit, ho w ev er, these ph ysical considerations p oint the wa y . W e need to let the sp eed of ligh t c go to ∞ , so one of the s cale p arameters should go to 0. In particular, we cann ot retain mo dular in v ariance. Accordingly , w e f ir st set a + = 2 π /µ, a − = ~ β , µ , ~ , β > 0 . (4.18) Here, we view β as 1 /mc , with m the particle mass, and w e trade a + for a parameter µ with dimension [p osition] − 1 to a void a great man y factors π . Next, the sp ectral v ariable y (dual p osition) is r ep laced b y the momen tum v ariable p = µy /β . (4.19) Finally , the coupling parameter b (with dimension [p osition]) is replaced b y b = g β , g > 0 , (4.20) so that g has dimension [action]=[p osition] × [momentum] as well as Planc k’s constan t ~ . With these changes in ef fect, it is easy to v erify the expansion A + ( b ; x ) = 2 + β 2 A + O  β 4  , β → 0 , (4.21) where A := − ~ 2 ∂ 2 x − g ~ µ coth ( µx/ 2) ∂ x − g 2 µ 2 / 4 , (4.22) and the limits lim β → 0 A − ( b ; x ) = exp( − iπ g / ~ ) T x 2 iπ /µ + ( i → − i ) =: M , x > 0 , (4.23) lim β → 0 A + ( b ; y ) = p − ig µ p T p i ~ µ + ( i → − i ) =: ˆ A, (4.24) A Relativi stic Conical F unction and its Whittak er Limits 27 while A − ( b ; y ) has no sen s ible limit. T he eig en v alue of A + ( b ; x ) on R ( b ; x, y ) satisf ies 2 c + ( y ) = 2 + β 2 p 2 / 4 + O  β 4  , (4.25) so that the eige n v alue of A b ecomes p 2 / 4, w h ile the eigen v alues of th e mono drom y op erator M and dual A∆O ˆ A r emain 2 cosh( π p/ ~ µ ) and 2 cosh( µx/ 2), resp. Lik ewise, the similarit y-transformed A∆O s A ± (2.45) and Hamiltonians H ± (2.69) yield A + ( b ; x ) , H + ( b ; x ) = 2 + β 2 H + O  β 4  , β → 0 , (4 .26) H := − ~ 2 ∂ 2 x + g ( g − ~ ) µ 2 4 sinh 2 ( µx/ 2) , (4.27) lim β → 0 A − ( b ; x ) = lim β → 0 H − ( b ; x ) = T x 2 iπ /µ + T x − 2 iπ /µ =: M , x > 0 , (4.28) lim β → 0 A + ( b ; y ) = T p i ~ µ + p + ig µ p T p − i ~ µ p − ig µ p =: ˆ A , (4. 29) lim β → 0 H + ( b ; y ) = X τ =+ , −  p − τ ig µ p  1 / 2 T p τ i ~ µ  p + τ ig µ p  1 / 2 =: ˆ H . (4.30) (W e tak e this opp ortunit y to p oin t out that the ‘ B C 1 v ersion’ of the mono drom y op erator (4.28 ) sp ecif ied in equation (8.20) of [10] con tains an er r or: the ph ase factor on the r.h.s. sh ou ld b e replaced b y 1.) The op erators A , M and H , M are related by a similarit y transformation with the limit function lim β → 0 φ ( b ) /c ( b ; x ) = w nr ( g / ~ ; µx/ 2) 1 / 2 , x > 0 , (4.3 1) where w nr ( λ ; r ) ≡ (2 sinh r ) 2 λ , λ, r > 0 , (4.32) the op erators ˆ A and ˆ A b y s im ilarity with lim β → 0 G ( ia − 2 ib ) G ( ib − ia ) /c ( b ; y ) = 2 / ˆ c nr ( g / ~ ; p/ ~ µ ) , ( 4.33) where the Harish -Chandra c -function is giv en b y ˆ c nr ( λ ; k ) ≡ 2Γ(2 λ ) Γ( λ ) Γ( ik ) Γ( λ + ik ) , (4.34) and the op erators ˆ A and ˆ H b y similarit y with ˆ w nr ( g / ~ ; p/ ~ µ ) 1 / 2 , where ˆ w nr ( λ ; k ) ≡ 1 / ˆ c nr ( λ ; ± k ) . (4.35) The limits (4.3 1) and (4.33) can b e readily v erif ied via the def inition (2.41) of the relativistic c -function and the ab o v e rep arametrizatio ns (4.18)–(4.20) by usin g the G -li mits (A.25)–(A.26). The functions (4.32) and (4.3 4) are normalized so that lim λ → 0 w nr ( λ ; r ) = 1 , lim λ → 0 ˆ c nr ( λ ; k ) = 1 . (4.36) Next, we s tu dy the nonrelativistic limit of the R -fun ction. T o th is end we f irst u se the def inition (1.1), the scaling prop ert y (2.19 ) and the limit lim t → 0 R ( π , t, t c ; v , tu ) = 2 F 1  ˆ c 0 + iu, ˆ c 0 − iu, c 0 + c 2 + 1 / 2; − sh 2 v  , (4.37) 28 S. Ruijsenaars ˆ c 0 ≡ ( c 0 + c 1 + c 2 + c 3 ) / 2 , (4.38) established and discussed in I. W e w rite the limit lim β → 0 R (2 π /µ, ~ β , g β ; x, β p /µ ) =: ψ nr ( g / ~ ; µx/ 2 , p/ ~ µ ) , (4.39) in terms of the dimensionless quan titie s λ ≡ g / ~ , r ≡ µx/ 2 , k ≡ p/ ~ µ, (4.40) already used in (4.32) and (4.34). Th e result reads ψ nr ( λ ; r , k ) = 2 F 1  ( λ + ik ) / 2 , ( λ − ik ) / 2 , λ + 1 / 2; − sin h 2 r  . (4.41) (See [27] for a limit that is related to (4.39), cf. (2. 24) .) Lik ewise, the alternativ e representat ion (2.14) en tai ls ψ nr ( λ ; r , k ) = 2 F 1  λ + ik , λ − ik , λ + 1 / 2; − sin h 2 ( r / 2)  , ( 4.42) whereas (2.15)–(2.16) again giv e rise to (4.41). The equalit y of (4.41) and (4.42) can b e rewritten as 2 F 1 ( a, b, a + b + 1 / 2; 4 w (1 − w )) = 2 F 1 (2 a, 2 b, a + b + 1 / 2; w ) , (4.43) whic h is a w ell-kno w n quadr atic transformation, cf. e.g. [28, p. 125]. Using other familiar features of the hypergeometric f unction, it is not dif f icult to verify that the op erators A , M and ˆ A do h a v e th e exp ected eigen v alues p 2 / 4, 2 cosh( p/ ~ µ ) and 2 cosh( µx/ 2) on the limit function ψ nr ( g / ~ ; µx/ 2 , p/ ~ µ ). More sp ecif ically , f or A this amoun ts to the ODE satisf ied b y 2 F 1 and for ˆ A this inv olves the cont iguous relations. The M -eigenfunction p rop erty follo ws b y using th e kn o wn analytic con tin uation of 2 F 1 ( a, b, c ; w ) across the logarithmic branc h cut w ∈ [1 , ∞ ). The ab o v e sp ecialization of the hypergeometric fun ction basically yields the so-called conical (or Mehler) fu nction. T o b e sp ecif ic, the latter can b e def in ed by P 1 / 2 − λ ik − 1 / 2 (cosh r ) ≡ (sinh r ) λ − 1 / 2 2 λ − 1 / 2 Γ( λ + 1 / 2) 2 F 1 ( λ + ik , λ − ik , λ + 1 / 2; (1 − cosh r ) / 2) , (4. 44) cf. [16, equation (14.3.15 )] and the hyp ergeometric function o ccurring her e equals the one in (4.4 2). W e n o w consid er the nonrelativistic limit of the m in imal representati ons of R ( b ; x, y ) derive d in Section 3. W e were n ot able to obtain a sensible limit for the second one, give n b y (3.48). F or the remaining four, ho w ev er, the limit can b e handled in a sense to b e explained s h ortly . F or exp ository reasons we f irst list the resulting represen tations for ψ nr ( λ ; r , k ): Γ(2 λ ) 2 λ Γ( λ ) 2 Z R dt 1 (cosh r + cosh t ) λ cos  k ln  cosh(( t + r ) / 2) cosh(( t − r ) / 2)  , (4.45) Γ(2 λ )(sinh r ) 1 − 2 λ 2 2 λ +1 π Z R dt Γ(( it − λ ± ik + c + 1) / 2) Γ(( it + λ ± ik + c + 1) / 2) exp(( i ( t − ic ) r ) , c > λ − 1 , (4.46) Γ(2 λ ) 4 π Γ( λ ) 2 Γ( λ ± ik ) Z R dt Γ(( it + λ ± ik ) / 2)Γ(( − it + λ ± ik ) / 2) exp ( itr ) , (4.47) Γ(2 λ ) 2 λ Γ( λ ± ik ) Z R dt exp( itk ) (cosh r + cosh t ) λ . (4.48) A Relativi stic Conical F unction and its Whittak er Limits 29 W e p ro ceed to discuss th ese formulas. First, w e note that th ey are deriv ed under the as- sumption λ, r , k ∈ (0 , ∞ ) , (4.49) and that this imp lies that th e int egrals in (4.45), (4.47) and (4.48) are ab s olutely con v ergen t. The in tegral in (4.46), how ev er, is only absolutely conv ergent for λ > 1 / 2; F or λ ∈ (0 , 1 / 2] it should b e view ed as a F ourier transform in the sense of temp ered distrib utions. Second, we compare these formulas to results in [16], where a h ost of r epresent ations for 2 F 1 and its conical function sp ecializat ion are listed. F ormula (4.48) can b e readily foun d there: It can b e obtai ned from equ ation (14 .12.4), whic h can b e written P 1 / 2 − λ ik − 1 / 2 (cosh r ) = r 2 π Γ( λ )(sinh r ) λ − 1 / 2 Γ( λ ± ik ) Z ∞ 0 dt cos k t (cosh r + cosh t ) λ . (4.50) (This in v olv es the dup lication formula of the gamma fun ction, equation (5.5) in [16].) As they stand, the three r emaining repr esen tations do n ot o ccur in [16]. Ho w ev er, as w as p oin ted out b y a referee, they can also b e tied in with results in th e v ast literature connected to the h yp ergeometric f u nction. T o b egin with, form ula (4.47) can b e derived (with some ef fort) b y com bining equations (15 .8.14) and (15.6.7 ) in [16]. It seems that th e formulas (4.45) and (4.46) cannot b e obtained by using only [16] or some other standard r eference b o ok. Ev en so, they agree with kno wn results. Ind eed, (4.45) amoun ts to equation (2.3) in the pap er [29], whereas (4.46) can b e derived by com bining sev eral sources. Sp ecif ically , the in tegral in (4.4 6) can b e view ed as a sp ecial case of the Mey er G -fu n ction, cf. Section 16.17 in [16] and p. 144, (2) of [30 ]. After con tour deformation, a residu e calcula tion leads to a formula in v olving a linear com bination of t w o 2 F 1 ’s with gamma fun ction co ef f icien ts, cf. equations (16.17 .2) and (16.17.3 ) in [16] or (7) in [30]. Finally , it f ollo ws from equation (3.2(27)) in [31] that the latter formula yields the conical function as represented by (4. 50). Third, none of the four represent ations (4.45)–(4.48 ) has b een obtained with complete r igor. The d iff icult y is to obtain uniform tail b oun d s on the p ertinen t integ rands that allo w an ap- plication of the dominated con v erge nce theorem. (In fact, to date a similar snag h as not y et b een obviate d for the limit trans ition (4.37) either.) In this connection we s h ould add that w e w ere u nable to v erify d ir ectly that eac h of the four form ulas giv es rise to a joint eigenfunction of the op erators A , M and ˆ A with eigen v alues p 2 / 4, 2 cosh ( p/ ~ µ ) and 2 cosh ( µx/ 2). O n the other hand, a direct c h ec k of the join t eigenfun ction prop erties of the f iv e relativistic represen- tations (3.47)–(3.5 1) seems not f easible either. W e con tin ue by sk etc hin g the deriv a tion of th e four formulas (4.45)–(4.48). First, we observe that an y factor of the form G ( a + , a − ; ia − ita − ) , (4.51) with t not dep end ing on a − , can b e treated via (A.25). Indeed, a scaling b y a + yields G (1 , κ ; i/ 2 + κ ( i/ 2 − it )) ∼ 2 π √ κ Γ( t ) exp ( t ln(2 π κ )) , κ → 0 . (4.52) In particular, th is y ields not only the asymptotics of the numerical prefactors, viz., 1 √ a + a − G ( ia − 2 ib ) ∼ Γ(2 λ ) exp(2 λ ln ( β ~ µ )) 2 π β ~ , β → 0 , (4.53) G ( ia − ib ) 2 √ a + a − G ( ia − 2 ib ) ∼ µ Γ(2 λ ) Γ( λ ) 2 , β → 0 , (4.54) 30 S. Ruijsenaars but also that of the y -dep end en t pr efactor in (3.50)–(3.51): G ( ia − ib ± y ) ∼ 2 π β ~ µ exp ( − 2 λ ln( β ~ µ )) Γ( λ ± ik ) , β → 0 . ( 4.55) T o handle the x -dep end en t prefactor in (3.48)–(3.49), h o w ev er, (A.25) is of no h elp and (A.26) seems not to apply either. But in fact w e can use the G -A∆Es (A.2) to f ir st write G ( x + ia − ib ) G ( x − ia + ib ) = 2 is − ( x − ib ) G ( x − ia + / 2 + ia − / 2 − ib ) G ( x − ia + / 2 − ia − / 2 + ib ) , (4.56) and then (A.26) can b e in v oked to ded uce the dominant asymptotics G ( ± x + ia − ib ) ∼ exp  π x β ~  (2 sinh r ) 1 − 2 λ , x > 0 , β → 0 . (4.57 ) The plane wa v es in (3.48)–(3.51) b ecome exp( iαz x ) = exp( iµz x/β ~ ) , exp( iαz y ) = exp( iz p/ ~ ) . (4.58) F or the f irs t and second plane w a ve w e now switc h to a new v ariable t giv en by z → β ~ t/ 2 , z → t/µ, (4.59) so that they tu rn into exp( itr ) and exp( itk ), resp . Lik ewise, in (3.47) w e change z to t/µ to get dimensionless G -function argu m en ts. With these v ariable c hanges in place , w e pro ceed to lo ok at the asymptotic b eha vior of the G -ratios in (3.47) –(3.51). F or the f irst case this is immediate from (A.26), and th is easily yields (4.45 ) when the p oin t wise limit is in terc hanged w ith the int egration. (As alluded to ab o v e, an L 1 -b ound un if orm f or β near 0 is needed to make th e interc h an ge r igorous. As w ell as in the next cases, no such b ound is av ailable for no w.) F or the second case (3.48), it seems not p ossible to get from the p oint wise b eha vior of the inte grand (with x > 0) as β goes to 0 a factor exp( − π x/β ~ ) that take s care of the dive rging factor exp( π x/β ~ ) coming from the prefactor (4.57). By contrast , for the third case w e may and will mak e a shift t → t + i/ (2 µβ ~ ) − ic, (4.60) with c ∈ R chosen so as to stay a wa y from p oles while shifting and taking β to 0. This cancels the div erging factor and results in (4.46) via (A.25). Finally , an application of (A.25) and (A.26) leads to the limits (4.47) and (4.48), resp. T o conclude th is sub section, w e p oint out that in view of (2.39) and (1.2 ) the n onrelativistic limit of E( b ; x, y ) is giv en by 2 w nr ( λ ; r ) 1 / 2 ˆ c nr ( λ ; k ) ψ nr ( λ ; r , k ) =: E nr ( λ ; r , k ) , (4.61) cf. (4.3 1)–(4.34 ). It has the unitary asym p totics E nr ( λ ; r , k ) ∼ exp( ir k ) − ˆ u nr ( λ ; − k ) exp( − ir k ) , r → ∞ , (4.62) with the scattering fun ction ˆ u nr ( λ ; k ) ≡ − ˆ c nr ( λ ; k ) ˆ c nr ( λ ; − k ) = − Γ( ik )Γ( λ − ik ) Γ( − ik )Γ( λ + ik ) . (4.63) A Relativi stic Conical F unction and its Whittak er Limits 31 The latter is normalized so that it equ als 1 for λ = 1, j ust as u ( b ; z ) (2.53) is normalized to equal 1 for b = a ± . I n this connection we w ould lik e to add that from (4.7) and (2.39)–(2.41) one readily d educes E( a ± ; x, y ) = 2 i sin( π xy /a + a − ) . (4.64) Hence the rep arametrizatio ns (4.18)–(4.20) and (4.40) yield E nr (1; r , k ) = 2 i sin k r. (4.65) Accordingly , the ‘fr ee’ theory with wh ic h the scattering is compared is giv en by the s ine transform (and not by the cosine transform, w hic h arises for b = λ = 0, cf. (2.58)). 5 The relativistic T o da case 5.1 T aking t he relativistic T o da limit Throughout this section, we require th at the parameters a + and a − b e p ositiv e. It is also con v enien t to r equire ( x, y ) ∈ (0 , ∞ ) 2 , (5.1) unt il further notice. In ke eping with our outline in the In tro duction, we b egin b y considering the b -v alues (1.16). In this case the w -fu nction (2.66) reads w ( a − iγ ; z ) = G ( ± z + ia ) /G ( ± z − γ ) , a = ( a + + a − ) / 2 , γ ∈ R , (5.2) and hence is no longer real-v alued for real z and γ 6 = 0. By contrast, the u -fu nction (2.53) is giv en by u ( a − iγ ; z ) = − G ( z ± γ ) /G ( z ± ia ) , (5. 3) so it is still u n itary for real z ; moreo v er, it is ev en in γ . Th e Hamiltonians (2.69) can b e wr itten H δ ( a − iγ ; z ) =  c δ ( z + ia − δ / 2 ± γ ) s δ ( z ) s δ ( z + ia − δ )  1 / 2 exp( ia − δ ∂ z ) + ( i → − i ) , (5.4) so they remain formally self-adjoin t for real z ; they are also eve n in γ . Next, we consider the f ive represen tations of the joint eigenfunction F( a − iγ ; x, y ) of the four Hamiltonians H ± ( a − iγ ; x ) and H ± ( a − iγ ; y ). Com bining (2.68) and (1.6) with (3.47)–(3.51), these are give n b y G ( − γ ) √ a + a −  G ( ± x + ia ) G ( ± x − γ ) G ( ± y + ia ) G ( ± y − γ )  1 / 2 Z R dz G ( z ± ( x − y ) / 2 − ia/ 2 − γ / 2) G ( z ± ( x + y ) / 2 + ia/ 2 + γ / 2) , (5.5) G ( − γ ) √ a + a −  G ( ± x + ia ) G ( ± x + γ ) G ( ± y + ia ) G ( ± y − γ )  1 / 2 Z R dz G ( ± z ± x / 2 − ia/ 2 + γ / 2) exp ( iαz y ) , (5.6) G ( γ ) √ a + a −  G ( ± x + ia ) G ( ± x + γ ) G ( ± y + ia ) G ( ± y − γ )  1 / 2 Z R dz G ( ± z ± y / 2 − ia/ 2 − γ / 2) exp ( iαz x ) , (5.7) G ( − γ ) √ a + a −  G ( ± x + ia ) G ( ± x − γ ) G ( ± y + ia ) G ( ± y + γ )  1 / 2 Z R dz G ( ± z ± y / 2 − ia/ 2 + γ / 2) exp ( iαz x ) , (5.8) G ( γ ) √ a + a −  G ( ± x + ia ) G ( ± x − γ ) G ( ± y + ia ) G ( ± y + γ )  1 / 2 Z R dz G ( ± z ± x / 2 − ia/ 2 − γ / 2) exp ( iαz y ) . (5.9) 32 S. Ruijsenaars (The square roots are p ositiv e for γ = 0.) As they s tand , none of these representa tions yields a manifestly real-v alued fu n ction for γ 6 = 0. Ho w ev er, comparing (5.7) and (5.8 ), w e see that these form ulae are related by a complex conjugatio n (tak e z → − z in one of them to chec k this). Lik ewise, (5.6) and (5.9 ) are related b y a complex conjugation. S ince the f iv e formulae yield the same function F( a − iγ ; x, y ), this function is in fact real-v alued. This r ealit y feature can b e tied to the b → 2 a − b sym metry of th e E-function, cf. (2.48 ) . In- deed, the latter in v ariance implies that E( a − iγ ; x, y ) is even in γ . No w since the u -fun ction (5.3) and th e phase φ ( a − iγ ) (giv en by (2.40)) are manifestly ev en in γ , it follo ws that F( a − iγ ; x, y ) is ev en in γ , cf. (2.64). C omparing once again (5.7) with (5.8) , and (5.6) with (5.9 ), we see that these form ulae are also related by f lipp ing the sign of γ , in accord with eve nness. Substituting γ → η + Λ , x → x + Λ , (5.10) w e are now prepared to study the T o d a limit Λ → ∞ . First, for r ∈ R w e hav e lim Λ →∞ c δ ( x + ir a − δ / 2 + η + 2Λ) c δ ( x + ir a − δ / 2 − η ) s δ ( x + Λ) s δ ( x + ir a − δ + Λ) = 1 + e δ ( − 2 x − ir a − δ + 2 η ) . (5.11) Th us we obtain relativistic nonp erio dic T o da Hamilto nians giv en b y H T δ ( η ; x ) ≡ lim Λ →∞ H δ ( a − iη − i Λ; x + Λ) = [1 + e δ ( − 2 x − ia − δ + 2 η )] 1 / 2 exp( ia − δ ∂ x ) + ( i → − i ) . (5.12) (The squ are ro ots are p ositiv e for x → ∞ .) Clearly , these are formally self-adjoin t on L 2 ( R , dx ). In this connection we p oint out that in view of th e dive rging x -shift, we ma y and will fr om no w on allo w x to v ary o v er R in the T o da quan tities, whereas we con tin ue to requ ir e that y b e p ositiv e. Next, w e n ote lim Λ →∞ e δ ( − 2Λ) c δ ( y + ir a − δ / 2 + η + Λ) c δ ( y + ir a − δ / 2 − η − Λ) = e δ (2 η ) / 4 . (5.13) Hence w e get dual Hamiltonia ns ˆ H T δ ( η ; y ) ≡ lim Λ →∞ e δ ( − Λ) H δ ( a − iη − i Λ; y ) = e δ ( η ) 2 s δ ( y ) − 1 / 2  exp( ia − δ ∂ y ) + exp( − ia − δ ∂ y )  s δ ( y ) − 1 / 2 , (5.14) whic h are formally p ositive on L 2 ((0 , ∞ ) , dy ). T o obtain the T o da limit of the joint eigenfunction F( a − iγ ; x, y ) in v olv es a great er ef fort. The k ey tool is the asymptotics (A.13) of the h yp erb olic gamma f unction. Th is enables us to sho w th at the limit F T ( η ; x, y ) ≡ lim Λ →∞ F( a − iη − i Λ; x + Λ , y ) (5.15) exists for eac h of the f iv e rep resen tations (5.5 )–(5.9). The detail s now follo w. T o start with, the asymptotic b eha vior for Λ → ∞ of the f ive p refactors can b e assem bled from the th r ee formulae G ( x + η + Λ) ∼ e − iχ exp  − iπ α 4  ( η + Λ) 2 + x 2 + 2 x ( η + Λ)   , (5.16) G ( ± y − η − Λ) ∼ e 2 iχ exp  iπ α 2  ( η + Λ) 2 + y 2   , (5.17) A Relativi stic Conical F unction and its Whittak er Limits 33 G ( ± ( x + Λ) + ia ) ∼ exp( π αa ( x + Λ)) . (5.18) Next, consider the in tegrand of (5.5) with the su b stitutions (5.10) . Two of the four G -functions are in v arian t, and the remaining t w o yield G ( z − ( x − y ) / 2 − ia/ 2 − η / 2 − Λ) G ( z + ( x + y ) / 2 + ia/ 2 + η / 2 + Λ) ∼ e 2 iχ exp  iπ α 2  [ z + y / 2] 2 + [( x + ia + η ) / 2 + Λ] 2   . (5.19) If w e no w co m bine the Λ-dep enden t terms coming fr om th e p refactor in (5.5), then w e see that they cancel the Λ-dep endent term in (5.19). The pr o duct of the r emainin g terms is r eadily v erif ied to b e giv en by  G ( ± y + ia ) a + a − G ( x − η )  1 / 2 exp(3 iχ/ 2) G ( z + ( x − y ) / 2 − ia/ 2 − η / 2) G ( z − ( x + y ) / 2 + ia/ 2 + η / 2) exp  iα 4 M  , (5.20) with M ≡ 2 z 2 + 2 z y − y 2 / 2 − iax − a 2 / 2 + iaη . (5.21) Finally , since we inte grate z in (5. 20) o v er R , we ma y sh ift z b y y / 2, yielding the limit function F T ( η ; x, y ) =  G ( ± y + ia ) a + a − G ( x − η )  1 / 2 exp(3 iχ/ 2) exp  iα 4  y 2 − ia ( x − η ) − a 2 / 2   × Z R dz G ( ± z + ( x − η − ia ) / 2) exp  iα 2  z 2 + 2 z y   . (5.22) T urn ing to th e integrand of (5.6 ), the s ubstitution (5.10 ) again lea v es t w o of the four G - functions un c hanged, as w ell as the plane w a v e factor. Th e remaining G -pr o duct has asymptotics G ( ± z + ( x − ia + η ) / 2 + Λ) ∼ e − 2 iχ exp  − iπ α 2  z 2 + [( x − ia + η ) / 2 + Λ] 2   . (5.23) Com bining this with the asymptotics of the p refactor follo wing from (5.16)–(5.18), the Λ- dep end en t terms drop out. T aking z → − z in the resulting limit function yields F T ( η ; x, y ) =  G ( ± y + ia ) a + a − G ( − x + η )  1 / 2 exp( − 3 iχ/ 2) exp  − iα 4  y 2 + ia ( x − η ) − a 2 / 2   × Z R dz G ( ± z − ( x − η + ia ) / 2) exp  − iα 2  z 2 + 2 z y   . (5.24) Comparing this representati on to (5.22), we see that eac h of the factors on the righ t-hand side is m atched by its complex-co njugate. Thus, the equalit y of (5.22) and (5.24) is in k eeping w ith the real-v aluedness of F T ( η ; x, y ) follo wing from its b eing th e limit of a real-v alued fun ction. Pro ceeding in th e same wa y for (5.9), we obtain as its limit again (5.22). O f course this should b e exp ected, since the factors on the right-hand side of (5.9) and (5.6) are r elated b y complex conjugation. On the other hand, the equalit y of the limits of (5.5) and (5.9) yields a non trivial chec k of the substan tial limit calculations. A t face v alue, the represen tations (5.7) and (5.8) seem not to giv e rise to a sensible T o da limit. In fact, h o w ev er , they do, bu t it is exp edient to p ostp one the d etails. First, we rewr ite th e 34 S. Ruijsenaars represent ations (5.22) and (5.24) in a more telling f orm, by brin ging in the G -cousins G L and G R , cf. (A.27)–(A.2 8). Ind eed, a straigh tforw ard calculatio n yields the equiv alen t represen tations F T ( η ; x, y ) =  G ( ± y + ia ) a + a − G R ( x − η )  1 / 2 e iαy 2 / 4 Z R dz G R ( ± z + ( x − η ) / 2 − ia/ 2) e iαz y , (5.25 ) and F T ( η ; x, y ) =  G ( ± y + ia ) a + a − G L ( η − x )  1 / 2 e − iαy 2 / 4 Z R dz G L ( ± z − ( x − η ) / 2 − ia/ 2) e iαz y . (5.26) (Here, the squ are ro ots are p ositiv e for x → ∞ , cf. (A.31).) No w w e use once ag ain the Planc h erel relation, as enco d ed in (3.22)–(3.23). T aking f irst f ( p ) = G R ( p + ( x − η ) / 2 − ia/ 2) exp( iα [ p + ( x − η + ia ) / 2][ y / 2 + ia/ 4]) , (5.27) g ( p ) = G R ( − p + ( x − η ) / 2 − ia/ 2) exp ( iα [ − p + ( x − η + ia ) / 2][ − y / 2 + ia/ 4]) , (5.28) w e can u s e (C.45) with s = a/ 4 to compute the F ourier transforms . This yields ˆ f ( q ) =  2 π α  1 / 2 e − iπ / 4 − 2 iχ exp  − iα 2 q ( x − η + ia )  G L ( q + y / 2 − 3 ia/ 4) , (5.29) ˆ g ( q ) =  2 π α  1 / 2 e − iπ / 4 − 2 iχ exp  iα 2 q ( x − η + ia )  G L ( − q − y / 2 − 3 ia/ 4) . (5.3 0) The in teg ral in (5.25) is therefore equal to exp  − iα 4 ( ia )( x − η + ia )  α 2 π Z R ˆ f ( q ) ˆ g ( − q ) dq = e − iπ / 2 − 4 iχ Z R dq exp  − iα ( q + ia/ 4)( x − η + ia )  G L ( q ± y / 2 − 3 ia/ 4) . (5.31) The new r ep resen tation thus obtained can b e s omewh at simplif ied b y rev erting to the G - function, and b y shifting the conto ur d o wn by a/ 4 (r ecall y > 0). In this wa y we obtain from (5.2 5) an alternativ e represen tation F T ( η ; x, y ) =  G ( ± y + ia ) a + a − G R ( x − η )  1 / 2 exp  iαy 2 / 8  × Z R + i 0 dz G ( z ± y / 2 − ia ) exp  − iα [ z ( x − η ) + z 2 / 2]  . (5.32) Pro ceeding in the same w a y for (5.26), w e arriv e at a f ourth represen tation, namely F T ( η ; x, y ) =  G ( ± y + ia ) a + a − G L ( η − x )  1 / 2 exp  − iαy 2 / 8  × Z R + i 0 dz G ( z ± y / 2 − ia ) exp  − iα [ z ( x − η ) − z 2 / 2]  . (5.33) Comparing it to (5.32 ) , w e deduce once again real-v aluedness of the fu nction F T ( η ; x, y ) on R 2 × (0 , ∞ ). Ha ving these t w o new rep r esen tations at hand, we can see w ith hind sigh t that they can also b e obtained from (5.8) and (5.7 ), resp ectiv ely . Indeed, when we let z → z + ia/ 2 − η / 2 − Λ / 2 (5.34) in the integ rand of (5.8), then w e obtain the t wo G -fun ctions featuring in (5.32), times t w o Λ-dep endent ones. If we n o w pro ceed in the same wa y as b efore, using (5.16)–(5.18 ) to handle the asymp totics of the p refactor, then w e arriv e once more at (5.32), yielding a c hec k on the rather extensiv e calculations. L ikewise, (5.7) giv es rise to (5.33). A Relativi stic Conical F unction and its Whittak er Limits 35 5.2 Asymptotic and analytic prop erties of F T ( η ; x , y ) With the v arious rep r esen tations of the function F T ( η ; x, y ) at our disp osal, sev eral salien t fea- tures can b e r eadily der ived. First, it is remark ably easy to show from (5.22) th at it has exp onenti al deca y for x → −∞ (as m igh t b e exp ected from the exp onentia l dive rgence of the ‘p oten tial’ factors in the Hamiltonians H T ± ( η ; x ) (5.12)). T o b e sp ecif ic, w e h av e a b ound F T ( η ; x, y ) = O (exp( αax/ 4)) , x → −∞ . (5.35) Insp ecting (5.22), it is clear that w e need only sh o w that the integ ral yields a function that is O (1) f or x → −∞ . T o this end we p oin t out that from (A.13 ) we ha v e estimates G ( v − ia/ 2) = O (exp( ∓ αav / 4) , v → ±∞ , (5.36) and that no p oles arise for r eal v . Hence the function v 7→ G ( v − ia/ 2) is b ounded on R . If w e no w tak e z → z + ( x − η ) / 2 in (5.22), then w e obtain a factor G ( z − ia/ 2) times a factor that is b ound ed for x, y , z , η ∈ R . Th us w e can in v ok e the b ound (5.36) on the f irst factor to dedu ce that the integral is in fact b ounded for x , y , η v arying ov er R , completing the pro of of (5.35). It is also not hard to obtain the x → ∞ asymptotics. T o this end w e start from th e repr esen- tation (5.32) and follo w th e reasoning b elo w (3.52). T h us w e shift the conto ur do wn by ǫ , where ǫ > 0 is small enough so that only the simp le p oles at z = ± y / 2 are passed (reca ll our stand ing assumption y > 0). The residues of the in teg ral th en follo w f rom (A.19), yielding a residue sum  G ( ± y + ia ) G R ( x − η )  1 / 2  G ( − y − ia ) exp ( iαy ( x − η ) / 2) + G ( y − ia ) exp( − iαy ( x − η ) / 2)  . (5.37) Using the G -a symptotics (A.13), it is easily verif ied that the remainder in tegral v anishes for x → ∞ , so that w e dedu ce F T ( η ; x, y ) ∼ u T ( η ; y ) 1 / 2 exp( iαxy / 2) + u T ( η ; − y ) 1 / 2 exp( − iαxy / 2) , x → ∞ . (5.38) Here w e hav e introdu ced the T o da u -function u T ( η ; y ) ≡ exp( − iαη y ) G ( − y + ia ) /G ( y + ia ) , (5.39) whic h can also b e written u T ( η ; y ) = c T ( η ; y ) /c T ( η ; − y ) , (5.40) with the T o da c -function def in ed by c T ( η ; y ) ≡ exp( − iαη y / 2) /G ( y + ia ) . (5.41) The corresp onding weigh t function is giv en b y w T ( y ) ≡ 1 /c T ( η ; ± y ) = G ( ± y + ia ) = 4 s + ( y ) s − ( y ) , (5.42) where w e us ed the G -A∆Es (A.2) in the last step. The dual counterparts of th ese formulae are not ob vious. T o b egin with, w e ha v e b een u nable to establish the large- y asymptotics of F T ( η ; x, y ). W e conjecture, ho w ev er, that th is is giv en b y F T ( η ; x, y ) ∼ e − iχ/ 2 − iπ / 8 G R ( x − η ) 1 / 2 exp( iα [ y 2 / 4 + ( x − η ) y / 2]) + e iχ/ 2+ iπ / 8 G R ( x − η ) − 1 / 2 exp( − iα [ y 2 / 4 + ( x − η ) y / 2]) , y → ∞ . (?) (5.43) 36 S. Ruijsenaars Ev en when this can b e sho wn, it is not clear whether the function G R ( x − η ) can b e view ed as an S -matrix f or the d ual dynamics. I n deed, the du al scattering theory seems quite unusual, just as at the classica l lev el [17]. Moreo ver, lik e the w -function w ( a − iη − i Λ; x + Λ), the u -function u ( a − iη − i Λ; x + Λ) has no limit for Λ → ∞ . On the other hand, the sim ilarity transforms A T δ ( η ; x ) ≡ G R ( x − η ) − 1 / 2 H T δ ( η ; x ) G R ( x − η ) 1 / 2 = exp( − ia − δ ∂ x ) + [1 + e δ ( − 2 x − ia − δ + 2 η )] exp ( ia − δ ∂ x ) , (5.44) can also b e obtained as the limits A T δ ( η ; x ) = lim Λ →∞ A δ ( a − iη − i Λ; x + Λ) , (5.45) cf. (2.4 5) and (5.1 1) . Note that they ha v e holomorphic co ef f icients, whereas the du al A∆Os ˆ A T δ ( η ; y ) ≡ w T ( y ) − 1 / 2 ˆ H T δ ( η ; y ) w T ( y ) 1 / 2 = ie δ ( η ) 2 s δ ( y )  exp( ia − δ ∂ y ) − exp( − ia − δ ∂ y )  , (5.46) (whic h are the coun terparts of A δ ( b ; y )), ha v e meromorphic co ef f icien ts. Surp risingly , the A∆Os A ± ( a − iη − i Λ; y ) ha v e no limit, wher eas they d o ha v e obvi ous T o da coun terparts, namely ˆ A T δ ( η ; y ) ≡ c T ( η ; y ) − 1 ˆ A T δ ( η ; y ) c T ( η ; y ) = exp( − ia − δ ∂ y ) + e δ ( η ) 2 s δ ( y ) exp( ia − δ ∂ y ) e δ ( η ) 2 s δ ( y ) . (5.47) Ev en though w ( a − iη − i Λ; x + Λ) has no limit either, there exists a function w T ( η ; x ) ≡ 1 /E ( ± ( x − η )) , (5.48) that ma y b e view ed as a weigh t fun ction. Here, E ( x ) is the fu n ction featurin g in (A.2 0), whic h w e already had o ccasion to us e in Section 3, cf. the paragraph con tai ning (3. 33). T o explain th is in terpretation, w e f irs t inv oke (A.22): T his r epresent ation m akes clear that w T ( η ; x ) is a real-a nalytic p ositiv e function on R . Secondly , w e note that when we set c T ( η ; x ) ≡ E ( x − η ) , (5.49) then w e get u T ( η ; x ) ≡ c T ( η ; x ) /c T ( η ; − x ) = G ( x − η ) . (5.50) Hence, ignoring ph ases and quadratic exp onentia ls, this u -fun ction enco des the conjectured large- y asymptotics (5.43) . Finally , the similarity- transformed A∆Os A T δ ( x − η ) ≡ w T ( η ; x ) − 1 / 2 H T δ ( η ; x ) w T ( η ; x ) 1 / 2 , (5.51) ha v e holomorphic co ef f icien ts. Sp ecif ically , from (A.23) w e compute A T δ ( x ) √ 2 π = e δ ( − x − ia − δ / 2) exp( − i ( x + ia − δ / 2) K − δ ) Γ( − i ( x + ia − δ / 2) /a δ + 1 / 2) exp( ia − δ ∂ x ) + ( i → − i ) . (5.52) Th us far, we ha v e kept x real and y p ositiv e in the function F T ( η ; x, y ). W e pr o ceed to stu dy its analytici t y features. T o this end, consider the function H ( x − η , y ) ≡ √ a + a − w T ( η ; x ) − 1 / 2 w T ( y ) − 1 / 2 F T ( η ; x, y ) . (5.53) A Relativi stic Conical F unction and its Whittak er Limits 37 The w eigh t fu nctions w T and w T (giv en by (5.42) and (5.48)) are well understo o d from an analytic viewp oin t, so w e need only clarify the c haracter of H ( x, y ). The f irst p oin t to note is that for eac h of the four in tegral rep r esen tations (5.22), (5.2 4) , (5.32) and (5.33) of F T ( η ; x, y ) the w eigh t function factors on the righ t-hand side of (5.53) ensur e th at the prefactors of th e z -integ rals b ecome ent ire functions of x and y . In deed, this is clear from the co rresp onding represent ations H ( x, y ) = E ( − x ) exp (3 iχ/ 2) exp  iα 4  y 2 − iax − a 2 / 2   × Z R dz G ( ± z + x / 2 − ia/ 2) exp ( iα [ z y + z 2 / 2]) , (5.54) H ( x, y ) = E ( x ) exp ( − 3 iχ/ 2) exp  − iα 4  y 2 + iax − a 2 / 2   × Z R dz G ( ± z − x / 2 − ia/ 2) exp  − iα [ z y + z 2 / 2]  , (5.55) H ( x, y ) = E ( − x ) exp ( − iχ/ 2) exp( iα ( y 2 − x 2 ) / 8) × Z R + i 0 dz G ( z ± y / 2 − ia ) exp  − iα [ z x + z 2 / 2]  , (5.56) H ( x, y ) = E ( x ) exp ( iχ/ 2) exp  iα ( x 2 − y 2 ) / 8  × Z R + i 0 dz G ( z ± y / 2 − ia ) exp  − iα [ z x − z 2 / 2]  . (5.57) W e ha v e sin gled out the function H ( x, y ), b ecause it extends fr om th e real x -axis and the p ositiv e y -axis (where it tak es real v alues) to a holomorphic (i.e., en tire) fu nction in x and y . T aking this assertion f or gran ted, the analytic characte r of F T ( η ; x, y ) can b e read of f from (5.53). W e con tin ue by pro ving the holomorphy claim. Consider f ir st the fu nction def ined b y the integral in (5.54). The in tegrand is a meromorph ic function I ( z ), wh ose asymptotics for Re z → ±∞ readily follo ws from the G -asymptotics (A.13). Sp ecif ically , we get I ( z ) = O (exp( − α Re z [ − I m x/ 2 + a/ 2 + Im z + Im y ]) , Re z → ∞ , (5.58) I ( z ) = O (exp( α Re z [ − Im x/ 2 + a/ 2 − Im z − Im y ]) , Re z → −∞ . (5.59) Therefore, exp on ential d eca y for Re z → ∞ can b e ac hiev ed b y taking Im z > Im x/ 2 − a/ 2 − Im y , (5.60) and for Re z → −∞ by taking Im z < − Im x/ 2 + a/ 2 − Im y . (5.61) Since the t wo G -functions d o not d ep end on y , this already implies th at H ( x, y ) extends to a h olomorph ic fu nction of y . Ind eed, when we con tin ue y of f the p ositiv e axis, we need only mo v e the cont our R up on the right and do wn on the left (when ever need b e) so as to r etain exp onenti al deca y . F or the x -con tin uation there is also no problem coming from the tail ends of the con tour, but w e n eed to a v oid that the con tour gets p in c hed b et w een the u p w ard and down w ard p ole sequences (cf. (A.16 )–(A.17)), z = − x/ 2 − ia/ 2 − z k l , z = x/ 2 + ia/ 2 + z k l , k, l ∈ N , (5.62) 38 S. Ruijsenaars as x is con tin ued of f the real axis. T h is can b e ac hiev ed b y requir ing x / ∈ − i [ a, ∞ ) . (5.63) As a result, H ( x, y ) extends to a holomorphic function of x and y outside the half line (5.63). T urn ing to the repr esen tation (5.55), w e can argue in the same w a y to conclude that exp o- nen tial deca y for Re z → ∞ can b e ac hieved b y taking Im z < Im x/ 2 + a/ 2 − Im y , (5.64) and for Re z → −∞ by taking Im z > − Im x/ 2 − a/ 2 − Im y . (5.65) Here w e get p oles for z = x/ 2 − ia/ 2 − z k l , z = − x/ 2 + ia/ 2 + z k l , k, l ∈ N , (5.66) as x is con tin ued of f the real axis. T hus we should require x / ∈ i [ a, ∞ ) , (5.67) so as to a void con tour pinc hing. It therefore follo ws that H ( x, y ) extends to a holomorphic function outside th e h alf line (5.67). Com bining these t w o conclusions, w e d ed uce that H ( x, y ) extends to a holomorph ic function on C 2 , as asserted. It also follo w s that the con tour in tegral in (5.54) extends to a meromorph ic function of x and y , w ith p oles only at x = − ia − z k l . Lik ewise, the con tour integ ral in (5.55) yields a meromorph ic fu nction w ith p oles only at x = ia + z k l . T o conclude th is accoun t of analyticit y features, we p oin t out that the holomorphy of H ( x, y ) can also b e derived in a somewhat dif ferent wa y f r om the t w o r epresent ations (5.56) –(5.57). F or these cases the t wo down w ard p ole sequen ces at z = ∓ y / 2 − z k l can alwa ys b e a voided by mo ving the con tour up. No w, how ever, another t ype of restriction arises from the requiremen t of exp onenti al d eca y on the cont our tails. F or th e inte grand in (5.56) w e n eed Im z < a/ 2 − Im x/ 2 on the righ t tail, but to obtain exp on ential d eca y on the left tail w e must r equire Im x > − a . Th us w e can only d educe holomorph y f or Im x > − a . Likewise, f or (5.57 ) we need Im x < a and Im z < a/ 2 + Im x / 2 on the left tail , so w e can only infer holomorphy for Im x < a . Ev en so, from these tw o f in dings w e can again conclud e holomorph y on C 2 . Moreo v er, it follo ws that th e con tour in tegral s in (5.56) and (5.57) give rise to meromorphic functions of x and y with p oles only at x = − ia − z k l and x = ia + z k l , r esp ectiv ely . 5.3 Join t eigenfunction prop erties T o complete this sectio n, w e v erify that the joint eigenfunction prop erties h a v e survived the Λ → ∞ limit. Due to the simpler analyticit y prop erties of the p ertinent con to ur integ rals (com- pared to the h yp erb olic case), this is rather straigh tforw ard. First, to sho w that F T ( η ; x, y ) is an eigenfunction of the Hamiltonians H T ± ( η ; x ) (5.12) with eigen v alues 2 c ± ( y ), we need only sho w that the A∆Os A T ± (0; x ) (5.44) ha v e the latter eigenv alues on the function G R ( x ) − 1 / 2 F T (0; x, y ). T o th is end we in v ok e the r epresent ation (5.26) . I t follo w s from our analysis of the conto ur in tegral in (5.55) that the con tour in teg ral in (5.26) with η = 0 def ines a f unction M ( x, y ) that extends to a meromorphic function of x and y with p oles o ccurring solely for x = ia + z k l . W e ma y write this function for x / ∈ i [ a, ∞ ) as M ( x, y ) = Z C dz K T ( x, z ) exp ( iαz y ) , (5.68) A Relativi stic Conical F unction and its Whittak er Limits 39 where w e hav e introdu ced K T ( x, z ) ≡ G L ( ± z − x/ 2 − ia/ 2) . (5.69) Also, the c hoice of Im z on the horizont al tails of th e con tour C dep ends on x an d y via (5.64)/(5.65) on th e right/l eft tail, while the tails are connected by a curve separating the upw ard and do wn w ard p ole s equ ences (5.66). No w our task is to p ro v e A T δ (0; x ) M ( x, y ) = 2 c δ ( y ) M ( x, y ) . (5.70) By virtu e of the already known meromorphy p rop erties, it suf f ices to sho w (5.70) for x v arying o v er a rectangle Re x ∈ ( − a, a ), Im x ∈ ( − 4 a, − 3 a ) (sa y), while k eeping y p ositiv e. The restric- tion on Im x ensur es that the upw ard and do wn w ard z -p ole sequences (5.66) are at a distance at least 2 a f r om th e real axis. Accordingly , we choose the con tour C to coi ncide with the real axis for Re z ∈ [ − 2 a, 2 a ] (say). F u rthermore, w e need to push do wn the righ t tail end and push u p the left tail end su f f icien tly far to retain exp onen tial deca y when we act with the x -shifts u n der the in tegral sign. The crux is no w that we h a v e a k ernel iden tit y K T ( x − ia − δ , z ) + [1 + e δ ( − 2 x − ia − δ )] K T ( x + ia − δ , z ) = K T ( x, z − ia − δ / 2) + K T ( x, z + ia − δ / 2) . (5.71) (This identit y can b e readily c hec ked by dividing f irst b y K T ( x, z − ia − δ / 2) and then usin g th e G L -A∆Es (A.30 ).) Th us we obtain A T δ (0; x ) Z C dz K T ( x, z ) e iαz y = Z C dz  K T ( x, z − ia − δ / 2) + K T ( x, z + ia − δ / 2)  e iαz y . (5.72) Shifting C up and do wn b y a − δ / 2, no p oles are met, and on th e resulting con tour s C + and C − there is still exp onentia l deca y at the left and r igh t. Hence we get e δ ( − y ) Z C + dz K T ( x, z ) e iαz y + e δ ( y ) Z C − dz K T ( x, z ) e iαz y . (5.73) Since the t wo integrands are now equal and the integrals yield the same v alue M ( x, y ), the eigen v alue prop erty (5.70) follo ws. W e p oin t out that the kernel identi t y (5.71) pla ys a role similar to the k ernel ident it y (3.14) of the hyp erb olic case. In the T o da case, how ever, w e can en sure that the z -p oles sta y at an arbitrary distance fr om th e real axis b y choosing Im x appropr iately , by con trast to the v -p oles in (3.15), cf. (3.16). Moreo v er, in the T o d a case the d eca y prop erties on the horizon tal tails of the con tour dep end on Im z , whereas the c hoice of Im z is irrelev ant in the h yp erb olic case (since the ‘ z 2 -terms’ d rop out in the | Re z | → ∞ asymptotics). Next, we n ote that to prov e that F T ( η ; x, y ) is a join t eig enfunction of the dual Hamilto- nians ˆ H T ± ( η ; y ) (5.14) with eigenv alues e ± ( x ), we need only s h o w that the A∆Os ˆ A T ± (0; y ) (5.46) ha v e these eigenv alues on the function w T ( y ) − 1 / 2 F T (0; x, y ). T o this en d w e us e the represen tation (5.32 ). Accordingly w e in trod uce ˆ M ( x, y ) = Z ˆ C dz ˆ K T ( x, z ) exp ( − iαz x ) , (5.74) where ˆ K T ( y , z ) ≡ G ( z ± y / 2 − ia ) exp( iα ( y 2 / 8 − z 2 / 2)) . (5. 75) 40 S. Ruijsenaars Recalling our analysis of the con tour integ ral in (5.56), we s ee that we sh ould require f irst of all Im x > − a in (5.74). T h en the inte gral is w ell def in ed when we choose the horizon tal tails of ˆ C equal to R (sa y ) on the left and ha ving Im z < a/ 2 − Im x/ 2 on th e righ t, w hile the middle part is ab o v e th e p ole sequences at z = ± y / 2 − z k l . F urthermore, the function ˆ M ( x, y ) is h olomorphic in x and y f or Im x > − a and extends to a meromorphic fu nction w ith p oles at x = − ia − z k l . In view of these features, w e need only p ro v e ˆ A T δ (0; y ) ˆ M ( x, y ) = e δ ( x ) ˆ M ( x, y ) , ( 5.76) for y v ary in g ov er a square Re y ∈ ( − a, a ), Im y ∈ ( − a, a ) (sa y ), w hile keeping x real. T o do so, w e choose the middle part of the con tour ˆ C equal to 2 ia + ( − 2 a, 2 a ), and connect this p art to ( −∞ , − 3 a ) and (3 a, ∞ ) in the obvious wa y . Th en the y -shifts can b e tak en under the integral sign w ithout z -p oles h itting the con tour. With these analytic preliminaries in place, the key algebraic p oint is once more a kernel iden tit y , namely , i 2 s δ ( y )  ˆ K T ( y + ia − δ , z ) − ˆ K T ( y − ia − δ , z )  = ˆ K T ( y , z − ia − δ / 2) . (5.77) (T o chec k it, one need only divide by th e r .h .s. and use the G -A∆Es (A.2).) Th us the l.h.s. of (5.76) equ als Z ˆ C dz ˆ K T ( x, z − ia − δ / 2) exp( − 2 iπ z x/a + a − ) . (5.78) Shifting the contour up b y a − δ / 2, no p oles are met, and s o th e joint eigenv alue equations (5.76) follo w. W e conclude th is s u bsection w ith s ome remarks. It follo ws from the eigen v alue features just prov ed that the holomorphic fun ction H ( x, y ) is a join t eigenfun ction of th e four A∆ O s A T ± ( x ) (5.52) and ˆ A T ± (0; y ) (5.46) with eigen v alues 2 c ± ( y ) and e ± ( x ), resp ectiv ely . The co ef f icien ts of the former A∆Os are entire, wh ereas the co ef f icien ts of the latter are mero- morphic. The co ef f icients of the A∆Os A T ± (0; x ) (5.44) are en tire as we ll, but th eir join t eigen- function G R ( x ) − 1 / 2 F T (0; x, y ) is meromorphic in x , with p oles for x = ia + z k l . This sho ws by example that the meromorphic vs. entire c haracter of th e co ef f icient s of the t yp e of comm uting A∆O p airs at issue in this pap er is compatible b oth with en tire and with meromorphic joint eigenfunctions. In this conn ection, it should b e noted that when the ratio a + /a − is irrational, then the only m ultipliers that do no destro y the join t eigenfunction prop erty are the constan ts. Another consequence w orth p oin ting out consists of the relations H ( x, k a δ + ia − δ ) = H ( x, k a δ − ia − δ ) , ∀ ( x, k , δ ) ∈ C × Z × { + , −} . (5.79) Indeed, these f ollo w f rom the eigen v alue equations H ( x, y − ia − δ ) − H ( x, y + ia − δ ) = 2 is δ ( y ) e δ ( x ) H ( x, y ) . (5.80) 6 The nonrelativistic T o da case In this section w e obtain the nonrelativistic coun terparts of the quantitie s in Section 5, along the lines laid out in S ubsection 4.2 for the hyp erb olic case. Th us w e switc h to the parameters (4.18) and momen tum v ariable (4.19), whereas the T o da analog of (4.20 ) is the substitution η = 2 µ ln( β µg ) , g > 0 . (6.1) A Relativi stic Conical F unction and its Whittak er Limits 41 F or the op erators H T + ( η ; x ) (5.12) and A T + ( η ; x ) (5.44), these sub stitutions en tail the expansion H T + , A T + = 2 + β 2 H + O  β 4  , β → 0 , (6.2) where H ≡ − ~ 2 ∂ 2 x + µ 2 g 2 exp( − µx ) , (6.3) is the nonr elativistic T o da Hamiltonian. Moreo v er, w e clearly get lim β → 0 H T − = lim β → 0 A T − = exp  2 iπ µ − 1 ∂ x  + ( i → − i ) =: M . (6.4) In view of (4. 25) the eigen v alue of the Hamiltonian H b ecomes p 2 / 4, while the eigen v alue of the mono dromy op erator M remains 2 cosh ( π p/ ~ µ ). T urn ing to the dual op erators, for ˆ H T + ( η ; y ) (5.14 ), ˆ A T + ( η ; y ) (5.46 ) and ˆ A T + ( η ; y ) (5.47 ) w e obtain (with p > 0) lim β → 0 ˆ H T + = µg p − 1 / 2  exp( i ~ µ∂ p ) + ( i → − i )  p − 1 / 2 =: ˆ H , (6.5) lim β → 0 ˆ A T + = iµg p − 1  exp( i ~ µ∂ p ) − ( i → − i )  =: ˆ A, (6. 6) lim β → 0 ˆ A T + = exp( − i ~ µ∂ p ) + µg p exp( i ~ µ∂ p ) µg p =: ˆ A , (6.7) and their eigen v alue b ecomes exp( µx/ 2). Th e du al operators ˆ H T − , ˆ A T − and ˆ A T − ha v e no limits, ho w ev er. Th e op erators ˆ A and ˆ H are related by a similarit y transformation with the square ro ot of the limit function lim β → 0 ( ~ β µ ) − 1 w T (2 π /µ, ~ β ; β p /µ ) = ˆ w nr ( p/ ~ µ ) , (6.8) where ˆ w nr ( k ) ≡ 2 k sinh( π k ) = 2 π / Γ( ± ik ) , ( 6.9) and the op erators ˆ A and ˆ A b y similarit y with lim β → 0 ( ~ β µ ) 1 / 2 c T (2 π /µ, ~ β , 2 µ − 1 ln( β µg ); β p/µ ) = ˆ c nr ( g / ~ ; p/ ~ µ ) , (6.10) where ˆ c nr ( λ ; k ) ≡ (2 π ) − 1 / 2 exp( − ik ln λ )Γ( ik ) . (6.11) (The limit (6. 8) is clear fr om (5.42), whereas (6. 10) follo ws b y using (A.2 5).) W e pr o ceed to obtain the n onrelativistic limit of the joint eige nfunction F T ( η ; x, y ) for eac h of the repr esen tatio ns (5.25), (5.26), (5.3 2) and (5.33), taking x real and y p ositiv e. First, w e h av e G R ( x − η ) → G R  2 π µ , ~ β ; x + 2 µ ln  ~ µg  + 2 µ ln  1 ~ β  . (6.12) Comparing this to the limit (A.33), w e see that it app lies to the β → 0 limit of (6.12), w ith th e parameter λ equal to 2. Th er efore, the limit of G R ( x − η ) equals 1, a circumstance that also explains wh y we get coinciding limits for H T δ and A T δ in (6.2)–(6. 4) , cf. (5.44). 42 S. Ruijsenaars Next, from (6.8) we see that the ab ov e su b stitutions imply lim β → 0 G ( ± y + ia ) a + a − = µp π ~ sinh( π p/ ~ µ ) . (6 .13) Moreo v er, αy 2 clearly v anishes for β → 0, so it no w follo ws th at the limits of the prefactors of the four repr esen tatio ns are all equal to the square ro ot of the righ t-hand s ide of (6.13) . T urn ing to the in tegrand in (5.25), the plane wa ve exp( iαz y ) b ecomes exp( iz p/ ~ ). F u rther- more, the sub stitutions on the t w o G R -functions yield G R  2 π µ , ~ β ; ± z + x 2 − iπ 2 µ − i ~ β 4 + 1 µ ln  ~ µg  + 1 µ ln  1 ~ β  . (6.14) Comparing once more to the limit (A.33), we see that it no w applies to the β → 0 limit of (6.1 4) with λ equ al to 1. Therefore, a short calc ulation giv es the limit exp  − 2 g ~ − 1 exp( − µx/ 2) cosh ( µz )  . (6.15) T aking z → t/µ in the inte gral, w e wind up with the limit function F T nr ( g / ~ ; µx/ 2 , p/ ~ µ ) , (6.16) where F T nr ( λ ; r , k ) = 2  π − 1 k sin h π k  1 / 2 Z ∞ 0 dt exp( − 2 λe − r cosh t ) cos ( tk ) . (6.17) It is readily ve rif ied that (5.26) also leads to the representat ion (6.17 ) for the limit function. Pro ceeding with the ab o v e subs titutions for (5.3 2), we see that w e can only get con v ergence of the exp onen tials for β → 0 when w e f irst r eplace the integrat ion v ariable z by ~ β w (sa y). Hence w e get a factor ~ β up fr on t, and the plane wa ve exp( − iαzx ) b ecomes exp( − iµwx ); moreov er, the quadratic exp onen tial con verges to 1 for β → 0. It th erefore remains to consider the r atio ~ β exp(2 iw ln( β µg )) /G  2 π µ , ~ β ; − ~ β w ± β p 2 µ + i 2  2 π µ + ~ β  . (6.18) Scaling the G -fun ctions b y µ/ 2 π , we can in v ok e (A.25 ) to deduce that (6.18 ) has β → 0 limit (2 π µ ) − 1 exp(2 iw ln( g/ ~ ))Γ( − iw ± ip/ 2 ~ µ ) . (6.19) Putting the pieces together, w e n o w get the limit function (6.16) repr esen ted as F T nr ( λ ; r , k ) = 1 4 π  π − 1 k sin h π k  1 / 2 Z R + i 0 dt Γ( i ( − t ± k ) / 2) exp ( it (ln λ − r )) . (6.20) The represent ation (5.33) also leads to (6.2 0). T h us w e obtain tw o d if ferent r epresen tations for the limit function (6.16). The resu lting identit y Z ∞ 0 dt exp  − 2 e − v cosh t  cos( tk ) = 1 8 π Z R + i 0 dt Γ( i ( − t ± k ) / 2) exp ( − itv ) , (6.21) is known and n ot hard to verify . Indeed, the eig enfunction transform asso ciated with (6.16) amoun ts to the Kont oro vic h –Leb edev tran s form, and the integ rals yield distinct representa tions of the mo d if ied Bessel fun ction K ik (2 e − v ), cf. e.g. [16, 10.3 2.9, 10.32.13 and 10 .43(v)]. F rom (6.17) it is ob vious that ˆ w nr ( k ) − 1 / 2 F T nr ( λ ; r , k ) extends from the p ositiv e k -axis to an en tire fun ction of k . Neither from (6.17) n or f rom (6.20) en tireness in r is manifest, bu t this is well A Relativi stic Conical F unction and its Whittak er Limits 43 kno wn (it can b e inferr ed, e.g., from ODE theory). As alrea dy men tioned in the In trod uction, the eigenfunction p rop erty f or the du al op erators seems not to o ccur in the standard sources. It is most ea sily c hec ked for ˆ A (6.6) b y pr o ceeding as in the relativistic case. Here it follo ws from the k ernel id en tit y ik − 1  ˆ K nr ( k + i, t ) − ˆ K nr ( k − i, t )  = ˆ K nr ( k , t − i ) , (6.22) where ˆ K nr ( k , t ) ≡ Γ( − it/ 2 ± ik / 2) . (6.23) (This iden tit y is the coun terpart of (5.77) .) The large- r asymptotics of F T nr ( λ ; r , k ) is wel l kno wn. It is giv en b y F T nr ( λ ; r , k ) ∼ ˆ u nr ( λ ; k ) 1 / 2 e ir k + ˆ u nr ( λ ; − k ) 1 / 2 e − ir k , r → ∞ , (6.24) where ˆ u nr ( λ ; k ) = ˆ c nr ( λ ; k ) / ˆ c nr ( λ ; − k ) , (6.25 ) and readily v erif ied from (6.20). It s eems m uc h h arder to obtain the large- k asymptotics from the ab o v e representat ions. Assuming plane-w a v e b ehavio r, a consider ation of the dual op erators leads to the exp ectation F T nr ( λ ; r , k ) ∼ exp( iφ + ik ln k − ik ) e i ( r − l n λ ) k + exp( − iφ − ik ln k + ik ) e − i ( r − l n λ ) k , k → ∞ , (6. 26) where φ ∈ [ − π , π ). Indeed, just like for its relativistic coun terpart (5.43 ), th is seems the simplest b ehavio r that is consisten t with the eigen v alues and dual eigen v alues. T o b e sure, f or neither case it is a priori clear that the asymptotic b eha vior must inv ol v e plane w a v es. A t an y rate, a r esult p ertinen t to (6.26) can b e found in th e literature: An asymptotic expansion for K ip ( x ) with p > x > 0 occurs in [32, Sectio n 7.13 .2, form ula (19)]. The dominant asymptotics d o es giv e r ise to (6.2 6) with φ = − π / 4, b ut an u nsettling O ( x − 1 ) error term is present . (If dep endence on x is included, then one wo uld rather exp ect an O ( x ) error term. Indeed, x → 0 corresp onds to v → ∞ , a limit for wh ich th e T o da p oten tial is exp onen tially v anishin g.) A The hyp erb olic gamma fu nction The hyperb olic gamma f unction w as int ro du ced and studied in [33] as a so-c alled minimal solu- tion of a sp ecial f irst ord er analytic dif feren ce equatio n. It is b asically the same as Ku rok aw a’s double sine [34], F addeev’s quan tum dilogarithm [35], and W oronowicz ’s quantum exp on ential function [36]. (The precise connectio ns b et w een these functions are sp elled out in App endix A of our p ap er [37].) In this app endix we review features of the hyp erb olic gamma function G ( a + , a − ; z ) that are used in the present pap er; if need b e, see [33] for p ro ofs. Unless sp ecif ied otherwise, we choose a + , a − > 0 , (A.1) and suppress the dep endence of G on a + , a − . T o b egin with, G ( z ) can b e def ined as the un ique minimal solution of one of the t w o analytic dif ference equations G ( z + ia δ / 2) G ( z − ia δ / 2) = 2 c − δ ( z ) , δ = + , − , (A.2) 44 S. Ruijsenaars that has mo dulus 1 for real z and s atisf ies G (0) = 1 (recall (1.9) for the n otation used here); remark ably , this en tails that the other one is then satisf ied as well. It is meromorphic in z , an d for z in the strip S ≡ { z ∈ C | | Im ( z ) | < a } , (A.3) no p oles and zeros o ccur. Hence w e h av e G ( z ) = exp( ig ( z )) , z ∈ S, (A.4) with the fu nction g ( z ) b eing holomorphic in S . Exp licitly , g ( z ) has the integral represen tation g ( a + , a − ; z ) = Z ∞ 0 dy y  sin 2 y z 2 sinh( a + y ) s inh( a − y ) − z a + a − y  , z ∈ S. (A.5) F rom this, the follo wing prop erties of the h yp erb olic gamma fu nction are immediate: G ( − z ) = 1 /G ( z ) , (ref lection equation) , (A.6) G ( a − , a + ; z ) = G ( a + , a − ; z ) , (mo dular in v ariance) , (A.7) G ( λa + , λa − ; λz ) = G ( a + , a − ; z ) , λ ∈ (0 , ∞ ) , (scale in v ariance) , (A.8) G ( a + , a − ; z ) = G ( a + , a − ; − z ) . (A.9) W e ha v e o ccasion to u se a few more features that are less ob vious, includ ing the duplication form ula G ( a + , a − ; 2 z ) = G ( a + , a − ; z ± ia + / 4 ± ia − / 4) , (A.10) the closely r elated form ula G ( a + , 2 a − ; 2 z ) = G ( a + , a − ; z ± ia + / 4) , (A.11) the explicit ev aluat ion G ( ia + / 2 − ia − / 2) = ( a + /a − ) 1 / 2 , (A.12) and the asymptotic b ehavior of G ( z ) for Re ( z ) → ±∞ . The latter is giv en b y G ( a + , a − ; z ) = exp  ∓ i  χ + αz 2 / 4   1 + O (exp ( − r | Re ( z ) | ))  , Re ( z ) → ±∞ , (A.13) where the d eca y rate r can b e an y p ositiv e n um b er satisfying r < α min( a + , a − ) , (A.14) and where χ ≡ π 24  a + a − + a − a +  . (A.15) Def in ing z k l ≡ ik a + + ila − , k , l ∈ N ≡ { 0 , 1 , 2 , . . . } , (A.16) the h yp erb olic gamma function h as its p oles at z = z − k l , z − k l ≡ − ia − z k l , k, l ∈ N , ( G -poles) , (A.17) A Relativi stic Conical F unction and its Whittak er Limits 45 and its zeros at z = z + k l , z + k l ≡ ia + z k l , k, l ∈ N , ( G -z eros) . ( A.18) The p ole at − ia is simp le and has residue lim z →− ia ( z + ia ) G ( z ) = i 2 π ( a + a − ) 1 / 2 . (A.1 9) In view of these features, G ( z ) can b e written as a ratio of entire fun ctions, G ( a + , a − ; z ) = E ( a + , a − ; z ) /E ( a + , a − ; − z ) , (A.20) where E ( a + , a − ; z ) has its zeros at z = z + k l , k, l ∈ N , ( E -zeros) . (A.2 1) The fu nction E ( a + , a − ; z ) w e ha v e o ccasion to emplo y is def ined in App endix A of I; it is closely r elated to Barnes’ double ga mma function [38]. W e need t w o more of its prop erties. First, from equations (A.41) and (A.43) in I w e hav e E ( z ) E ( − z ) = exp  1 2 Z ∞ 0 dy y  1 − cos(2 y z ) sinh( a + y ) s inh( a − y ) − z 2 a + a −  e − 2 a + y + e − 2 a − y   , (A.22) where z b elongs to the strip S (A.3). Second, we need the A∆Es it satisf ies, namely , E ( z + ia δ / 2) E ( z − ia δ / 2) = √ 2 π exp( iz K δ ) / Γ( iz /a − δ + 1 / 2) , δ = + , − , (A.23) K δ ≡ 1 2 a − δ ln  a δ a − δ  , (A.24) cf. equations (A.46)–(A.47) in I. W e also state tw o zero step size limits of the hyp erb olic gamma fu nction, wh ic h we n eed for taking nonrelativistic limits. T he f irst one yields the relation to the Euler gamma f unction: lim κ ↓ 0 G (1 , κ ; κz + i/ 2) exp  iz ln(2 π κ ) − ln(2 π ) / 2  = 1 / Γ( iz + 1 / 2) . (A.25) F or the second on e w e need to require that z s ta y aw ay from cuts given b y ± i [ a + / 2 , ∞ ). Then w e ha v e lim a − ↓ 0 G ( a + , a − ; z + iua − ) G ( a + , a − ; z + ida − ) = exp(( u − d ) ln[2 c + ( z )]) , u, d ∈ R , (A.26) uniformly on compact subsets of the cut plane. F or the relativistic T o da setting it is exp edien t to emplo y t w o slightl y dif ferent h yp erb olic gamma f unctions def ined b y G R ( z ) ≡ exp( ig R ( z )) , g R ( z ) ≡ g ( z ) + χ + αz 2 / 4 , (A.27) G L ( z ) ≡ exp( ig L ( z )) , g L ( z ) ≡ g ( z ) − χ − αz 2 / 4 . (A.28) These functions are the unique minimal solutions of the analytic dif ference equations G R ( z + ia − δ / 2) G R ( z − ia − δ / 2) = 1 + e δ ( − 2 z ) , δ = + , − , ( A.29) 46 S. Ruijsenaars G L ( z + ia − δ / 2) G L ( z − ia − δ / 2) = 1 + e δ (2 z ) , δ = + , − , (A.30) with asymptotic b eha vior G R L ( z ) = 1 + O ( exp( − r | Re ( z ) | )) , Re ( z ) → ±∞ . (A.31) F urthermore, they are r elated by G R ( z ) G L ( − z ) = 1 . (A.32) The prop erties of the functions G R and G L just stated are easy to infer from the corresp onding prop erties of the h yp erb olic gamma fu n ction. (In App endix A of [37] w e already in trod u ced functions S R and S L that dif fer from G R and G L b y the sh if t z → z − ia .) Finally , w e ha v e occasion to u se the limits lim a − ↓ 0 g R L ( a + , a − ; z ± λs ( a + , a − )) =  ± a + 2 π e + ( ∓ 2 z ) , λ = 1 , 0 , λ > 1 , (A.33) where s ( a + , a − ) ≡ a + 2 π ln 1 a − , (A.34) whic h h old un iformly for z v arying o v er arbitrary compact sub sets of C . T o our kno wledge, these limits h a v e not b een obtained b efore. W e p resen t their pr o of in the next app endix. B Pro of of (A.33) Since we h a v e g L ( z ) = − g R ( − z ) , (B.1) w e need only sho w (A.33) for g R . Our pro of actually yields a stronger result, whic h m ay b e useful in other con texts. T o state th is resu lt, we f ix λ ∈  1 / √ 2 , ∞  , (B.2 ) and c h o ose δ satisfying δ ∈  1 , √ 2  , δλ > 1 . (B.3) Then we sh all sho w g R ( a + , a − ; z + λs ( a + , a − )) = a λ − 2 sin( π a − /a + ) e + ( − 2 z ) + O  a δλ − 1 −  , a − ↓ 0 , (B.4) where the implied constan t can b e c hosen uniformly for z v arying o v er compact sub sets of C . A k ey ingredien t of ou r pro of is the comparison fun ction w e us ed in Sub section I I I A of [33 ] to obtain the G -asymptotics (A.13). Sp ecif ically , w e f irs t fo cus on the dif ference d ( a + , a − ; z ) ≡ a + a − g R ( a + , a − ; z ) − A 2 g R ( A, A ; z ) , z ∈ S , (B.5) for the sp ecial A -c hoice A ≡  a 2 + + a 2 − 2  1 / 2 . (B.6) A Relativi stic Conical F unction and its Whittak er Limits 47 Observing that A ≥ a (with equalit y for a + = a − ), we deduce that d ( z ) is well def ined and holomorphic for z in the str ip S (A.3). Recalling (1.4), (A.15 ) and (A.27), w e see that we may rewrite d ( z ) as d ( a + , a − ; z ) = 1 2 i Z R dy I ( a + , a − ; y ) exp(2 iy z ) , (B.7) where I ( a + , a − ; y ) ≡ 1 2 y  a + a − sinh( a + y ) s inh( a − y ) − A 2 sinh 2 ( Ay )  . (B.8) The A -c hoice (B. 6) is th e unique one guaran teeing that I ( y ) has no p ole at y = 0. Sp ecif ically , w e easily calculate I ( y ) = c ( a + , a − ) y + O  y 3  , y → 0 , (B.9) where c ( a + , a − ) is a p olynomial in a + and a − of degree 4. Since w e let a − go to 0, w e m a y an d will assume from n o w on 1 A > δ 1 a + . (B.1 0) Next, w e shift the y -con tour up b y r ≡ π δ /a + . (B.11) On acco unt of (B.1 0), this ensu res that only the simple p ole at y = iπ /a + is passed. The residue at this p ole is readily calculate d, yieldin g the representati on d ( z ) = a + a − 2 sin( π a − /a + ) e + ( − 2 z ) + ρ ( z ) , z ∈ S, (B.12) where ρ is the remainder integral ρ ( z ) ≡ 1 2 i exp( − 2 r z ) Z R duI ( u + ir ) exp (2 iuz ) , r = π δ /a + , z ∈ S. (B.13) W e are no w pr ep ared to replace z by z + λs ( a + , a − ) , | Im z | ≤ a + / 2 , (B.14) so that we get, using (B.11), d ( z + λs ) = a + a − a λ − 2 sin( π a − /a + ) e + ( − 2 z ) + 1 2 i a δλ − e + ( − 2 δ z ) Z R duI ( u + iπ δ /a + ) exp(2 iu ( z + λs )) . (B.15) Next, w e note lim a − ↓ 0 I ( w ) = 1 2 w  a + w s in h( a + w ) − a 2 + / 2 sinh 2 ( a + w/ √ 2)  . ( B.16) Since z is required to satisfy | Im z | ≤ a + / 2, it follo ws that the u -integ rand in (B.15) remains b ound ed by a f ixed L 1 ( R )-function as a − ↓ 0. Thus we readily deduce the b ound d ( z + λs ) a + a − = a λ − 2 sin( π a − /a + ) e + ( − 2 z ) + O  a δλ − 1 −  , | Im z | ≤ a + / 2 , a − ↓ 0 , ( B.17) with the imp lied constant uniform on compact subsets of the strip | Im z | ≤ a + / 2. 48 S. Ruijsenaars Next, w e claim that w e h a v e A 2 a + a − g R ( A, A ; z + λs ) = O ( a δλ − 1 − ) , | Im z | ≤ a + / 2 , a − ↓ 0 , (B.18) uniformly on compacts of | Im z | ≤ a + / 2. T aking this claim for grant ed, we see from (B.5) and (B.17) that (B.4) holds true, uniformly on compacts of the latter strip. No w the A∆E (A.29) with δ = − implies g R ( a + , a − ; z + ia + / 2 + λs ) − g R ( a + , a − ; z − ia + / 2 + λs ) = − i ln  1 + exp  − 2 π a − ( z + λs )  . (B.19) Hence w e hav e (us in g the def in ition (A.34) of s ) g R ( z + ia + / 2 + λs ) − g R ( z − ia + / 2 + λs ) = O  exp  − a + a − ln 1 a −  , a − ↓ 0 , (B.20) uniformly on compacts of C . F r om this it is routine to in fer that (B.4) also holds f or z ∈ C , with the b ound u niform f or z v arying o v er arbitrary C -compacts. It remains to prov e the claim. T o th is end w e start from the iden tit y g R ( A, A ; z ) = 1 π b + ( π z / A ) , b + ( w ) ≡ Z ∞ w dt te − t sinh( t ) , (B.21) whic h follo ws from equations (3.41)– (3.46) in [33]. Next, w e n ote th e b oun d b + ( w + R ) = R exp( − 2( w + R ))(1 + O (1 /R )) , R → ∞ , (B.22) where the implied constan t can b e chosen uniform on C -compacts. (One can u s e the elemen tary in tegral Z ∞ x te − ct dt = 1 c 2 e − cx (1 + cx ) , c > 0 , (B.23) to v erify this estimate.) As a consequence, we obtain A 2 a + a − g R ( A, A ; z + λs ) ∼ λ 2 π √ 2 a λ √ 2 − 1 − ln  1 a −  exp( − 2 π √ 2 z /a + ) , z ∈ C , a − ↓ 0 , (B.24) uniformly f or z in C -compacts. Th us (a stronger v ersion of ) our claim follo ws. This concludes the pro of of (B.4), and so (A.33) follo ws as an ob vious corollary . C F o urier transform form ulas W e f ix complex num b ers µ , ν satisfying − a < Im µ < Im ν < a, a = ( a + + a − ) / 2 . (C.1) Hence w e hav e Im ( ν − µ ) ∈ (0 , a + + a − ) , (C.2) A Relativi stic Conical F unction and its Whittak er Limits 49 and the fun ction I ( µ, ν ; x ) ≡ G ( x − ν ) /G ( x − µ ) , (C.3) is p ole-free in the strip Im ν − a < Im x < Im µ + a. (C.4) Also, f rom the G -asymptotics (A.13) w e d educe I ( µ, ν ; x ) = O (exp( ∓ α Im ( ν − µ )Re x/ 2)) , Re x → ±∞ . (C.5) Therefore, for r eal y the function F ( µ, ν ; y ) ≡ Z R dx exp ( iαxy ) G ( x − ν ) G ( x − µ ) , (C.6) is wel l def in ed, and analytic in µ and ν in the r egion (C.1). Moreo ver, w e retain exp onent ial deca y of the in tegrand when we let y v ary o ver the s tr ip | Im y | < Im ( ν − µ ) / 2 , (C.7) so F ( µ, ν ; y ) is analytic in y in this strip and giv en b y (C.6). Our f irst aim is to obtain the F ourier transform (C.6 ) in explicit form. Since the func- tion I (C.3) has no p oles in the strip (C.4), we can mak e a con tour shift x → z + ( µ + ν ) / 2 , x, z ∈ R , (C.8) to deduce F ( µ, ν ; y ) = exp( iαy ( µ + ν ) / 2) Z R dz exp( iαz y ) G ( ± z − κ ) , (C.9 ) where w e hav e set κ ≡ ( ν − µ ) / 2 , Im κ ∈ (0 , a ) . (C.10) Hence calculating F amount s to f inding the cosine transform of G ( ± z − κ ) for κ in the s tr ip Im κ ∈ (0 , a ). T he r easoning in th e pro of of the follo wing prop osition, ho w ev er, hinges on staying at f irst with (C.6). A sp ecial case of the f ollo wing result amounts to a hyp erb olic analog of a trigonometric b eta in tegral in Raman ujan’s lost noteb o ok. More precisely , (C.11) with y = ( µ + ν ) / 2 amoun ts to equation (1.8) with τ < 0 in Stokman’s pap er [39]. F ormula (C.11) is also obtained in Chapter 5 of v an d e Bult’s Ph.D. thesis [40 ] (cf. Th eorem 5.6.8 with n = 1), as a result of sp ecializing more general formulas. In slightl y dif f er ent guises it occur r ed previously in v arious pap ers (the earliest ones b eing [41, 42, 43]), bu t a complete pro of cannot b e found there. Prop osition C.1. The F ourier tr ansfor m (C.6) adm its the explicit evaluation F ( µ, ν ; y ) = ( a + a − ) 1 / 2 exp( iαy ( µ + ν ) / 2) G ( ia − 2 κ ) G ( ± y − ia + κ ) , (C.11) wher e µ, ν satisfy (C.1) , y satisfies (C.7) , and κ is given by (C.10) . 50 S. Ruijsenaars Pro of . W e f ir st take y real and choose µ, ν such that Im ( ν − µ ) ∈ ( a l , a + + a − ) , a l ≡ m ax ( a + , a − ) . (C.12) This en tails that w e ca n shift ν do wn b y ia δ or µ up b y ia δ without lea ving the region (C.1). Hence w e may shift under the integral s ign and use the G -A∆Es (A.2) to obtain F ( µ, ν − ia δ ; y ) = 2 Z R dxe iαxy c − δ ( x − ν + ia δ / 2) G ( x − ν ) /G ( x − µ ) , y ∈ R . (C.13) Recalling that F ( µ, ν ; y ) is analyti c in y wh en (C.7) holds, this can b e rewritten as F ( µ, ν − ia δ ; y ) = e − δ ( − ν + ia δ / 2) F ( µ, ν ; y − ia δ / 2) + e − δ ( ν − ia δ / 2) F ( µ, ν ; y + ia δ / 2) . (C.14) Lik ewise, w e obtain F ( µ + ia δ , ν ; y ) = e − δ ( − µ − ia δ / 2) F ( µ, ν ; y − ia δ / 2) + e − δ ( µ + ia δ / 2) F ( µ, ν ; y + ia δ / 2) . (C.15) Next, consider th e con tour shift x → x + is , where s ∈ (Im ν − a, I m µ + a ) , (C.16) cf. (C.4). It implies F ( µ, ν ; y ) = Z R dx exp ( iα ( x + is ) y ) G ( x + is − ν ) /G ( x + is − µ ) = exp ( − αs y ) F ( µ − is, ν − is ; y ) , y ∈ R . (C.17) Since F ( µ, ν ; y ) is analytic in y in the strip (C.7), it now follo ws from (C.12) that we ha v e F ( µ − is, ν − is ; y ) = e αsy F ( µ, ν ; y ) , | Im y | ≤ a l / 2 . (C.18) No w in (C.15) we ma y tak e µ, ν → µ − is, ν − is with s ∈ (0 , I m µ + a ). Then we are entitle d to in v oke (C.18) to get F ( µ − is + ia δ , ν − is ; y ) = e − δ ( − µ + is − 3 ia δ / 2) e αsy F ( µ, ν ; y − ia δ / 2) + e − δ ( µ − is + 3 ia δ / 2) e αsy F ( µ, ν ; y + ia δ / 2) . (C.19) In this equation w e can let s con v er ge to a δ , whic h yields F ( µ, ν − ia δ ; y ) = e − δ (2 y − µ − ia δ / 2) F ( µ, ν ; y − ia δ / 2) + e − δ (2 y + µ + ia δ / 2) F ( µ, ν ; y + ia δ / 2) . (C.20) Comparing (C.20) and (C.14), we see that the dif ference yields a linear relation b et w een F ( µ, ν ; y + ia δ / 2) and F ( µ, ν ; y − ia δ / 2). After some simplif ication, this relation can b e rewritten as F ( µ, ν ; y + ia δ / 2) F ( µ, ν ; y − ia δ / 2) = e − δ ( − µ − ν ) c − δ ( y − ia + κ ) c − δ ( y + ia − κ ) . (C.21) In tro ducing F r ( µ, ν ; y ) ≡ exp( iα ( µ + ν ) y / 2) G ( ± y − ia + κ ) , (C.22) A Relativi stic Conical F unction and its Whittak er Limits 51 it is easy to v erify that F r also satisf ies this y -A∆E. Since we can c ho ose δ = + , − and a + /a − / ∈ Q , it readily follo ws that we must h a v e F ( µ, ν ; y ) = C ( µ, ν ) F r ( µ, ν ; y ) , (C.23) with C indep endent of y . The up shot of our reasoning thus far is that when µ and ν are restricted by (C.12), then the F ourier tr an s form is of the f orm (C.23), with F r giv en by (C.22). By analyticit y in µ and ν , this relation now extends to the whole region (C.1) and then, b y analyticit y in y , to the str ip (C.7). In view of (C.9), we th er efore hav e sh o wn Z R dz exp( iαz y ) G ( ± z − κ ) = C ( µ, ν ) G ( ± y − ia + κ ) , (C.24) Im κ ∈ (0 , a ) , | Im y | < Im κ. It follo ws fr om th is th at w e hav e C ( µ, ν ) = D ( µ − ν ) , (C.25) so it r emains to pro v e D ( µ − ν ) = ( a + a − ) 1 / 2 G ( ia + µ − ν ) . (C.26) T o this en d w e su b stitute F ( µ, ν ; y ) = D ( µ − ν ) exp( iα ( µ + ν ) y / 2) G ( ± y − ia + κ ) (C.27) in (C.14), and divide the result by D ( µ − ν ) exp( iα ( µ + ν ) y / 2) G ( ± y − ia δ / 2 − ia + κ ) . (C.28) Using the G -A∆Es, a straigh tforw ard calculation then yields D ( µ − ν + ia δ ) D ( µ − ν ) = 2 is − δ ( µ − ν + ia δ ) . (C.29) Setting D r ( µ − ν ) ≡ G ( µ − ν + ia ) , (C.30) w e see that D r also satisf ies the A∆E (C.29). T h us we d ed uce as b efore D ( µ − ν ) = η D r ( µ − ν ) , (C.31) where η can only dep end on a + and a − . As a result, we ha v e no w p ro v ed Z R dx exp( iαxy ) G ( ± x − κ ) = η G ( ia − 2 κ ) G ( ± y − ia + κ ) , (C.32) Im κ ∈ (0 , a ) , | Im y | < Im κ. T o calculate η , w e choose κ equal to ia − / 2. Using the G -A∆Es, this yields Z R dx exp( iαxy ) 1 c + ( x ) = η G ( ia + / 2 − ia − / 2) 1 c − ( y ) . (C.33) No w th e in tegral is elemen tary , yielding a + /c − ( y ). Using (A.12) w e then infer η equals ( a + a − ) 1 / 2 , hence completing the pro of.  52 S. Ruijsenaars In view of (C.32), the cosine trans form of G ( ± x − κ ) is p rop ortional to G ( ± y − i ˆ κ ), with ˆ κ ≡ ia − κ. (C.34) T o b e sp ecif ic, w e h a v e  2 α π  1 / 2 Z ∞ 0 cos( αxy ) G ( ± x − κ ) dx = G ( ˆ κ − κ ) G ( ± y − ˆ κ ) , (C.35) Im κ ∈ (0 , a ) , y ∈ R . Note that the pr op ortionalit y constan t can n o w b e c hec k ed b y taking the cosine trans form of (C.35). W e pro ceed by deriving a corollary of the ab ov e result, which w e n eed in S ubsection 5.1. First, we f ix ν in th e strip Im ν ∈ (0 , a ), so that G ( x − ν ) has exp onen tial deca y for x → ±∞ , and c h o ose µ r eal, so that G ( x − µ ) is a phase for real x . Consider no w the in tegral  α 2 π  1 / 2 Z R dx exp ( iαxy ) G ( x − ν )[exp( − iχ − iαµ 2 / 4 + iαµx/ 2) G ( − x + µ )] , (C.36) y ∈ R . In virtue of the G -asymptotics, the p h ase factor in square brac k ets conv erges to exp( iαx 2 / 4) for µ → −∞ . By dominated con v ergence, the integ ral therefore con v erge s to  α 2 π  1 / 2 Z R dx exp ( iαxy ) G ( x − ν ) exp( iαx 2 / 4) , y ∈ R , Im ν ∈ (0 , a ) . (C.37) On the other hand, b y (C.11) the int egral (C.36) equals exp( − iχ + iα [ y ( µ + ν ) / 2 + µν / 4]) G ( µ + ia − ν ) G ( µ + y + ia − ν / 2) G ( y − ia + ν / 2) . (C.38) Using once more the G -asymptotic s, this has a µ → −∞ limit th at can b e written as e − iπ / 4 − 4 iχ G ( y − ia + ν / 2) exp  − iα 4  ( y − ia + ν / 2) 2 + 4( y + ν / 2)( ia − ν ) + ν 2   . (C.39) The resulting equalit y of (C.37) and (C.39 ) can b e rewritten in a more illuminating w a y by using the G -cousins G R (A.27) an d G L (A.28). Indeed, let us set z ≡ y + ν / 2 , Im ν ∈ (0 , a ) , y ∈ R , (C.4 0) so that (C.37) and (C.39) can b e wr itten as exp( − iαν 2 / 4)  α 2 π  1 / 2 Z R dx exp( iαxz ) G ( x − ν ) exp  iα ( x − ν ) 2 / 4  , (C.41) and exp( − iαν 2 / 4) e − iπ / 4 − 3 iχ G L ( z − ia ) exp( − iαz ( ia − ν )) . (C.42) Hence w e hav e  α 2 π  1 / 2 Z R dx exp ( iα ( x + ia − ν ) z ) G ( x − ν ) exp  iα ( x − ν ) 2 / 4  = e − iπ / 4 − 3 iχ G L ( z − ia ) , (C.43) with z giv en by (C.40). C ho osing no w ν = 2 is , w e obtain the follo win g corollary . A Relativi stic Conical F unction and its Whittak er Limits 53 Corollary C.2. L etting w = x + ia − 2 is, z = y + is, s ∈ (0 , a/ 2) , x, y ∈ R , (C.44) we have  α 2 π  1 / 2 Z R exp( iαwz ) G R ( w − ia ) d Re w = exp( − iπ / 4 − 2 iχ ) G L ( z − ia ) , (C.45) wher e χ is gi v en by (A.15) . Th us w e hav e reco v ered the F ourier transform (A.21) in [37]. Ac kno wledgmen ts W e would lik e to thank M. 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