math-ph 2026-03-17 0

BC Toda chain II: symmetries. Dual picture

In the previous paper we derived Gauss-Givental integral representation for the wave functions of quantum BC Toda chain and also introduced Baxter operators for this model. In the present paper we prove commutativity of Baxter operators, as well as s…

Authors: N. Belousov, S. Derkachov, S. Khoroshkin

BC Toda chain II: symmetries. Dual picture
B C T o da c hain I I: symmetries. Dual picture N. Belouso v † , S. Derk ac ho v ⋄† , S. Khoroshkin ∗◦† † Beijing Institute of Mathematic al Scienc es and Applic ations, Huair ou district, Beijing, 101408, China ⋄ Steklov Mathematic al Institute, F ontanka 27, St. Petersbur g, 191023, Russia ∗ Dep artment of Mathematics, T e chnion, Haifa, Isr ael ◦ Skolkovo Institute of Scienc e and T e chnolo gy, Skolkovo, 121205, R ussia Abstract In the previous pap er w e deriv ed Gauss–Given tal integral represen tation for the wa ve functions of quan tum B C T oda chain and also introduced Baxter op erators for this mo del. In the present paper we pro ve commutativit y of Baxter op erators, as w ell as show that the constructed w av e functions are symmetric with resp ect to signed p ermutations of sp ectral parameters and diagonalize Baxter op erators. F urthermore, we deriv e Mellin–Barnes in tegral represen tation for the wa ve functions. With its help we show that w av e functions satisfy dual system of difference equations with resp ect to sp ectral parameters and coincide with h yp ero ctahedral Whittak er functions. Finally , w e give heuristic pro ofs of orthogonalit y and completeness of the wa v e functions. Con ten ts 1 In tro duction 2 1.1 Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 W a ve functions and Baxter operators . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Gauss–Giv en tal represen tation 12 2.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Raising op erator and reflection symmetry . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Baxter op erator and its reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Comm utativity and exchange relations . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Prop erties of wa v e functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Lo cal relation b etw een GL and B C op erators . . . . . . . . . . . . . . . . . . . . 27 2.7 Scalar pro duct b etw een GL and B C wa ve functions . . . . . . . . . . . . . . . . 31 2.8 F rom Gauss–Given tal to Mellin–Barnes . . . . . . . . . . . . . . . . . . . . . . . 34 2.9 Orthogonalit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Dual system 39 3.1 GL system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.1 Dual Hamiltonians, raising and Baxter op erators . . . . . . . . . . . . . . 39 3.1.2 Lo cal relations and eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 41 3.2 B C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.1 Dual Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 GL – B C intert winers and Mellin–Barnes in tegrals . . . . . . . . . . . . . . 43 1 3.2.3 QISM tec hnique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.4 Asymptotics and h yp ero ctahedral Whittaker function . . . . . . . . . . . 54 3.2.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 A In tegral iden tities 59 A.1 Pro of of flip relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 A.2 Pro of of reduced flip relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 B Gauss–Giv en tal b ounds 61 B.1 GL system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 B.2 B C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 B.3 Kernels of op erator pro ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 B.4 F rom equalit y of kernels to equality of operators . . . . . . . . . . . . . . . . . . 68 B.5 Bounds on B C T o da wa v e function . . . . . . . . . . . . . . . . . . . . . . . . . . 68 C Mellin–Barnes b ounds and analyticit y 70 C.1 GL system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 C.2 B C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 D Gustafson in tegral reduction 74 1 In tro duction The quan tum T o da chain of B C n t yp e is go verned by the Hamiltonian [ Skl2 , I ] H B C = − n X j =1 ∂ 2 x j + 2 n − 1 X j =1 e x j − x j +1 + 2 α e − x 1 + β 2 e − 2 x 1 , (1.1) whic h acts on functions of n spatial v ariables x j ∈ R and con tains t w o parameters α , β . In [ BDK ] w e found the reflection op erator, which acts on functions of one v ariable and can b e used to diagonalize B C 1 T oda Hamiltonian. F urthermore, com bining reflection operator with Skly anin’s in tertwining op erators [ Skl3 ] w e constructed the mono dromy op erators satisfying reflection equa- tion asso ciated with B C n system. Reductions of mono dromy op erators pro duce the so-called raising and Baxter op erators. With the help of raising op erators we derived the explicit ex- pression for the wa ve function of B C n system in a form of Gauss–Given tal iterated in tegrals. Besides, we prov ed the comm utativity of Baxter op erators with B C n T oda Hamiltonians and Baxter equation c haracterising their sp ectra. The same technique w as used in [ ADV1 , DKM1 , BKP ] for the analysis of q -T oda chain and Heisen b erg spin chains, while for GL n T oda c hain ( α = β = 0) the analogous construction repro duces the kno wn results of [ GLO2 ] obtained using representation theory of classical Lie groups GL ( n, R ). In the presen t paper w e further study the w av e functions of B C n T oda c hain using other tools kno wn in the theory of quantum in tegrable systems. First, w e pro ve comm utativity and exchange relations for Baxter and raising op erators constructed in [ BDK ]. These relations corresp ond to certain integral identities, which can b e describ ed and derived in a graphical language using a few basic transformations called star-triangle and flip relations. In addition, we sho w that our raising and Baxter op erators, as well as their pro ducts, are well defined on the spaces of p olynomially b ounded contin uous functions. 2 Second, using the relations b et ween the Baxter and raising operators, we diagonalize the Baxter op erators and establish the symmetry of the wa ve functions with resp ect to signed p erm utations of the sp ectral parameters. Next, we compute the scalar pro duct b et ween GL n and B C n w av e functions. This computa- tion leads us to the Mellin–Barnes integral representation of B C n w av e functions, generalizing Iorgo v and Shadura results [ IS ] for B n system ( β = 0). F urther, we sho w that Gauss–Given tal and Mellin–Barnes integral representations are essen tially sufficient to prov e the orthogonalit y and completeness relations for the B C n w av e functions. It is kno wn that prop erly normalized wa ve functions of B C n T oda c hain, also called h yp ero c- tahedral Whittak er functions, enjoy difference equations in sp ectral parameters [ DE2 ]. T o v erify the dual equations for our wa v e functions we note that the kernel of Mellin–Barnes represen tation can b e regarded as Ruijsenaars B C n — GL n k ernel function, which satisfies corresp onding dif- ference equations. In addition, w e found the second kernel function and another Mellin–Barnes in tegral representation for B C n w av e functions. Finally , follo wing the strategy of [ HR , BCDK ] we establish asymptotics of our wa v e function and identify it with the hypero ctahedral Whittaker function by v an Diejen and Emsiz. The Mellin–Barnes iterativ e in tegral can b e calculated by residues and the result is the precise form ula for the corresp onding Harish-Chandra series. In the following subsections w e briefly recall the construction of comm uting Hamiltonians and their eigenfunctions for b oth GL n and B C n T oda c hains (see [ BDK ] and references therein for details), and then state the main results of this pap er. 1.1 Hamiltonians F or the description of quan tum T oda chains of GL and B C types we need t wo basic blo c ks: Lax matrix asso ciated with the j -th particle [ F , G , Skl1 ] L j ( u ) = u + i ∂ x j e − x j − e x j 0 ! (1.2) and the reflection (b oundary) matrix K ( u ) = − α u − i 2 − β 2  u − i 2  − α ! . (1.3) With their help w e construct mono drom y matrix T n ( u ) T n ( u ) = L n ( u ) · · · L 1 ( u ) = A n ( u ) B n ( u ) C n ( u ) D n ( u ) ! , (1.4) whic h serves for GL T o da chain, and mono dromy matrix T n ( u ) T n ( u ) = T n ( u ) K ( u ) σ 2 T t n ( − u ) σ 2 = A n ( u ) B n ( u ) C n ( u ) D n ( u ) ! , σ 2 = 0 − i i 0 ! , (1.5) 3 whic h desrib es B C T oda system (to distinguish ob jects related to B C system we often use the mathbb font). The elements A n ( u ) and B n ( u ) are generating functions for commuting Hamilto- nians of GL and B C T o da chains correspondingly A n ( u ) = u n + n X s =1 u n − s H s , [H s , H r ] = 0 , (1.6) B n ( u ) = ( − 1) n  u − i 2  u 2 n + n X s =1 u 2( n − s ) H s  , [ H s , H r ] = 0 . (1.7) The first co efficien ts are related to the quadratic Hamiltonian ( 1.1 ), namely , H 1 = − H B C and H 1 = n X j =1 i ∂ x j , H 2 = 1 2  H 2 1 − H GL  , (1.8) where H GL = H B C   α = β =0 = − n X j =1 ∂ 2 x j + 2 n − 1 X j =1 e x j − x j +1 . (1.9) Tw o other sp ecial cases of B C T o da system are B T o da chain ( α  = 0, β = 0) and C T o da c hain ( α = 0, β  = 0). 1.2 W av e functions and Baxter op erators Denote tuples of n v ariables and sums of their comp onen ts x n = ( x 1 , . . . , x n ) , x n = x 1 + . . . + x n . (1.10) The wa ve functions of GL and B C systems are parametrized by zeroes of the generating functions eigen v alues A n ( u ) Φ λ n ( x n ) = n Y j =1 ( u − λ j ) Φ λ n ( x n ) , (1.11) B n ( u ) Ψ λ n ( x n ) = ( − 1) n  u − i 2  n Y j =1 ( u 2 − λ 2 j ) Ψ λ n ( x n ) . (1.12) As sho wn in [ BDK ], they can b e constructed using the so-called raising op erators. T o in tro duce these op erators denote g = 1 2 + α β . (1.13) Besides, in what follo ws we alwa ys assume conditions g > 0 , β > 0 . (1.14) 4 The raising op erators are integral op erators acting on functions φ ( x n − 1 ) b y the formulas  Λ n ( λ ) φ  ( x n ) = Z R n − 1 d y n − 1 Λ λ ( x n | y n − 1 ) φ ( y n − 1 ) , (1.15)  V n ( λ ) φ  ( x n ) = Z R n − 1 d y n − 1 V λ ( x n | y n − 1 ) φ ( y n − 1 ) , (1.16) where the k ernels are given by Λ λ ( x n | y n − 1 ) = exp  i λ  x n − y n − 1  − n − 1 X j =1 ( e x j − y j + e y j − x j +1 )  , (1.17) V λ ( x n | y n − 1 ) = (2 β ) i λ Γ( g − i λ ) Z R n d z n (1 + β e − z 1 ) − i λ − g (1 − β e − z 1 ) − i λ + g − 1 θ ( z 1 − ln β ) × exp  i λ  x n + y n − 1 − 2 z n  − n − 1 X j =1 ( e z j − x j + e z j − y j + e x j − z j +1 + e y j − z j +1 ) − e z n − x n  . (1.18) Here and in what follo ws by θ ( x ) we denote the Heaviside step function θ ( x ) = ( 1 , x ≥ 0 , 0 , x < 0 . (1.19) The w av e functions are defined b y iterative integrals Φ λ n ( x n ) = Λ n ( λ n ) · · · Λ 1 ( λ 1 ) · 1 , (1.20) Ψ λ n ( x n ) = V n ( λ n ) · · · V 1 ( λ 1 ) · 1 , (1.21) whic h by tradition are called G auss–Giv ental represen tations. Note that from the explicit for- m ula ( 1.17 ) we hav e Λ n ( λ + ρ ) = e i ρ x n Λ n ( λ ) e − i ρ x n − 1 , (1.22) and consequen tly , Φ λ n + ρ e n ( x n ) = e i ρ x n Φ λ n ( x n ) , e n = (1 , . . . , 1) . (1.23) Example 1. In the case n = 1 we ha ve Φ λ ( x ) = e i λx and Ψ λ ( x ) = e x 2 √ 2 β W 1 2 − g , − i λ (2 β e − x ) = (2 β ) i λ Γ( g − i λ ) Z ∞ ln β dz e i λ ( x − 2 z ) − e z − x (1 + β e − z ) − i λ − g (1 − β e − z ) − i λ + g − 1 , (1.24) where W κ,µ ( z ) is the Whittak er function [ DLMF , Chapter 13 ]. The abov e in tegral representation for it is w ell known [ DLMF , (13.16.5) ]. 5 The close relativ es of the r aising operators are Baxter operators. These are integral op erators acting on functions φ ( x n ) b y the formulas  Q n ( λ ) φ  ( x n ) = Z R n d y n Q λ ( x n | y n ) φ ( y n ) , (1.25)  Q n ( λ ) φ  ( x n ) = Z R n d y n Q λ ( x n | y n ) φ ( y n ) , (1.26) where the k ernels are given by Q λ ( x n | y n ) = exp  i λ  x n − y n  − n − 1 X j =1 ( e x j − y j + e y j − x j +1 ) − e x n − y n  , (1.27) Q λ ( x n | y n ) = (2 β ) i λ Γ( g − i λ ) Z R n +1 d z n +1 (1 + β e − z 1 ) − i λ − g (1 − β e − z 1 ) − i λ + g − 1 θ ( z 1 − ln β ) × exp  i λ  x n + y n − 2 z n +1  − n X j =1 ( e z j − x j + e z j − y j + e x j − z j +1 + e y j − z j +1 )  . (1.28) These op erators commute with the Hamiltonians of the corresp onding systems  Q n ( λ ) , A n ( u )  = 0 ,  Q n ( λ ) , B n ( u )  = 0 , (1.29) and also satisfy Baxter equations Q n ( λ ) A n ( λ ) = i − n Q n ( λ − i) , Q n ( λ ) B n ( λ ) = − β ( g + i λ ) 2 λ Q n ( λ − i) , (1.30) see [ BDK ]. As shown in App endix B , all of the ab o v e in tegral op erators are w ell defined on the spaces of p olynomially b ounded contin uous functions P n =  φ ∈ C ( R n ) : | φ ( x n ) | ≤ P ( | x 1 | , . . . , | x n | ) , P — p olynomial  . (1.31) Namely , by Prop ositions B.1 , B.2 w e hav e Λ n ( λ ) : P n − 1 → P n , λ ∈ R , (1.32) V n ( λ ) : P n − 1 → P n , λ ∈ R , (1.33) Q n ( λ ) : P n → P n , Im λ < 0 , (1.34) Q n ( λ ) : P n → P n , Im λ ∈ ( − g , 0) . (1.35) As a result, the products of these op erators are w ell defined. In particular, the Gauss–Given tal represen tations ( 1.20 ), ( 1.21 ) conv erge and wa ve functions Φ λ n ( x n ) , Ψ λ n ( x n ) are p olynomially b ounded. In the previous pap er [ BDK ] we also pro ved that the w av e function Ψ λ n ( x n ) decays rapidly in the classically forbidden regions x j +1 ≪ x j , as well as x 1 ≪ 0, see Prop osition B.3 here for the precise statemen t. 6 Remark 1. In [ BDK ] all relations with op erators hold on the spaces of exp onen tially temp ered smo oth functions, whic h are suited for the action of mono drom y matrix entries. In this pap er w e work with the spaces P n that are more suitable for the pro ducts of in tegral op erators: note that the Baxter op erators do not preserve the space E n ( R n ) considered in [ BDK , Corollary 5]. 1.3 Main results In what follo ws we use shorthand notation for the pro ducts of gamma functions Γ( a ± b ) = Γ( a + b ) Γ( a − b ) , Γ( ± a ± b ) = Γ( a + b ) Γ( a − b ) Γ( − a + b ) Γ( − a − b ) . (1.36) In Section 2 using diagram tec hnique we deduce the following relations for B C raising and Baxter op erators. Prop ositions 2.1 , 2.3 . The r elations V n ( λ ) = V n ( − λ ) , λ ∈ R , V n +1 ( λ ) V n ( ρ ) = V n +1 ( ρ ) V n ( λ ) , λ, ρ ∈ R , Q n ( λ ) Q n ( ρ ) = Q n ( ρ ) Q n ( λ ) , Im λ, Im ρ ∈ ( − g , 0) , Q n ( λ ) V n ( ρ ) = Γ(i λ ± i ρ ) V n ( ρ ) Q n − 1 ( λ ) , Im λ ∈ ( − g , 0) , ρ ∈ R , hold on the sp ac es P n − 1 and P n c orr esp ondingly. In the last iden tity for n = 1 we denote Q 0 ( λ ) = (2 β ) − i λ Γ(2i λ ) Γ( g + i λ ) Id . (1.37) The ab o ve relations imply that the B C w a ve function ( 1.21 ) is the eigenfuction of Baxter op erators, symmetric with resp ect to the action of the W eyl group of the ro ot system B C n . Theorem 2.1 . L et σ ∈ S n , ε n ∈ { 1 , − 1 } n . Then Ψ λ 1 ,...,λ n ( x n ) = Ψ ε 1 λ σ (1) ,...,ε n λ σ ( n ) ( x n ) . (1.38) Theorem 2.2 . L et λ n ∈ R n and Im λ ∈ ( − g , 0) . Then Q n ( λ ) Ψ λ n ( x n ) = (2 β ) − i λ Γ(2i λ ) Γ( g + i λ ) n Y j =1 Γ(i λ ± i λ j ) Ψ λ n ( x n ) . Note that the ab o v e symmetry implies non-degeneracy of the Hamiltonians eigenv alues ( 1.12 ), while the last form ula is in accordance with the Baxter equation ( 1.30 ). Besides, for comparison, let us write the relations b et w een GL raising and Baxter op erators Λ n +1 ( λ ) Λ n ( ρ ) = Λ n +1 ( λ ) Λ n ( ρ ) , (1.39) Q n ( λ ) Q n ( ρ ) = Q n ( ρ ) Q n ( λ ) , (1.40) Q n ( λ ) Λ n ( ρ ) = Γ(i λ − i ρ ) Λ n ( ρ ) Q n − 1 ( λ ) , (1.41) 7 whic h lead to the analogous statements for the GL wa v e functions ( 1.20 ) Φ λ 1 ,...,λ n ( x n ) = Φ λ σ (1) ,...,λ σ ( n ) ( x n ) , σ ∈ S n , (1.42) Q n ( λ ) Φ λ n ( x n ) = n Y j =1 Γ(i λ − i λ j ) Φ λ n ( x n ) , (1.43) see [ GLO1 , Theorem 2.3]. With the help of the ab ov e Baxter op erators and diagram technique in Section 2.7 we cal- culate the follo wing scalar pro duct b et w een GL and B C wa ve functions. Prop osition 2.5 . L et λ n ∈ R n , γ n ∈ C n such that Im γ 1 = . . . = Im γ n < 0 . Then Z R n d x n Φ − γ n ( x n ) Ψ λ n ( x n ) exp( − β e − x 1 ) = (2 β ) − i γ n n Q j,k =1 Γ(i γ j ± i λ k ) Q 1 ≤ j 0 Ψ λ n ( x n ) = e β e − x 1 Z ( R − i ε ) n d γ n ˆ µ ( γ n ) (2 β ) − i γ n n Q j,k =1 Γ(i γ j ± i λ k ) Q 1 ≤ j 0. The chain relations from Figure 5 represent reductions of star-triangle relations as z → ±∞ , when some lines disapp ear. F or example, taking limit z → −∞ of the first star-triangle relation with the help of form ulas lim z →−∞ (1 − e z − w ) i λ − 1 θ ( w − z ) = 1 , lim z →−∞ exp( − e z − x ) = 1 (2.3) 13 x y λ = Γ(i λ ) x λ λ y λ x y − λ = Γ(i λ ) x λ y Figure 5: Chain relations w e arrive at the first chain relation. T o justify interc hange of limit and integration note that “c hain” integral is analytic in λ in the domain Re(i λ ) > 0, see Figure 3 . So, one can, at first, assume Re(i λ ) ≥ 1, then apply dominated con vergence theorem using the b ound   (1 − e z − w ) i λ − 1 θ ( w − z )   ≤ 1 , (2.4) and analytically con tinue the answer at the end. λ λ = λ λ λ − λ λ λ = λ − λ − λ λ = λ λ λ − λ Figure 6: Cross and reduced cross relations 14 The cross relation from Figure 6 can b e prov ed in tw o steps using star-triangle transfor- mations, as sho wn in Figure 7 . Let us remark that in these pictures and in what follows w e frequen tly omit lab els of v ertices. λ λ = 1 Γ(i λ ) λ = − λ λ λ λ λ − λ Figure 7: Pro of of cross relation using star-triangle relations Also note that b oth sides of cross relation are analytic in λ ∈ C . Th us, to prov e it using star-triangle relations w e assume Re(i λ ) > 0 and analytically contin ue the answ er at the end. The tw o other relations from Figure 6 can b e obtained from the first one in the limit when one of non-in tegrated v ariables tends to ±∞ . λ ρ − 2 λ − λ − ρ − 2 ρ y = λ − ρ x λ − ρ ρ λ − 2 ρ x − λ − ρ − 2 λ λ − ρ y λ − ρ Figure 8: Flip relation No w consider flip relation from Figure 8 . V ertical lines come with step functions, hence, diagram from the left represen ts expression (2 β ) i( λ + ρ ) Γ( g − i λ ) Γ( g − i ρ ) F ( x, y ) θ ( x − ln β ) θ ( y − ln β ) , (2.5) where w e denoted F ( x, y ) = e i( λ − ρ ) x − 2i ρy (1 + β e − y ) − i ρ − g (1 − β e − y ) − i ρ + g − 1 × Z x ln β dz e − 2i λz (1 + β e − z ) − i λ − g (1 − β e − z ) − i λ + g − 1 ( e z + e y ) i( λ + ρ ) (1 − e z − x ) i( λ − ρ ) − 1 . (2.6) The flip relation just states that this function is symmetric F ( x, y ) = F ( y , x ), the pro of is giv en in App endix A.1 . Let us note that the abov e in tegral is absolutely conv ergent under condition Re(i ρ ) < Re(i λ ) < g . F or calculations w e actually need not flip relation itself, but the transformation from Figure 9 . The latter follo ws from the sequence of flip and star-triangle relations pictured in Figure 10 . On the last step of this sequence t wo dashed lines cancel each other. 15 λ ρ λ ρ − λ − ρ λ − ρ ρ − λ − 2 λ − 2 ρ = ρ λ ρ λ − λ − ρ − 2 ρ − 2 λ Figure 9: Twin p eaks relation Diagrams from Figure 9 represent absolutely conv ergen t integrals under condition Re(i λ ), Re(i ρ ) < g , whic h we imp ose b ecause of double lines. T o p erform all steps from Figure 10 w e should also assume Re(i ρ ) < Re(i λ ). After calculations this assumption can b e remov ed b y analytic con tinuation. λ ρ λ ρ − λ − ρ λ − ρ ρ − λ − 2 λ − 2 ρ λ ρ ρ − λ ρ λ − ρ λ − ρ ρ − λ − ρ ρ − λ − 2 λ − 2 ρ ρ λ ρ − λ ρ ρ λ − ρ λ − ρ − λ − ρ ρ − λ − 2 ρ − 2 λ ρ λ ρ λ − λ − ρ − 2 ρ − 2 λ Figure 10: Sequence of flip and star-triangle relations x λ y = ( e x + e y ) − i λ = 1 λ − λ Figure 11: Cancellation of lines 16 λ − 2 λ − λ − ρ ln β = Γ(i λ − i ρ ) λ − ρ x λ − ρ ρ − 2 ρ x − λ − ρ ln β Figure 12: Reduced flip relation In the limit y → ln β + flip relation reduces to the in tegral identit y pictured in Figure 12 . This limiting identit y is pro ven in App endix A.2 . The further limit x → + ∞ is straightforw ard and giv es the relation depicted in Figure 13 . λ − 2 λ − λ − ρ ln β = (2 β ) i ρ Γ(i λ − i ρ ) Γ( g − i ρ ) Figure 13: Relation from Figure 12 in the limit x → + ∞ A combination of reduced flip and star-triangle relations gives another useful transformation sho wn in Figure 14 . Due to presence of double lines it holds under assumption Re(i λ ) , Re(i ρ ) < g . Moreo ver, in the case ρ = − λ one of the dashed lines disapp ears and the identit y simplifies, see Figure 15 . λ − ρ λ − λ − ρ ln β = λ − ρ − 2 λ ρ − λ ρ − λ − ρ ln β − 2 ρ Figure 14: Combination of reduced flip and star-triangle relations λ λ λ 2 λ − 2 λ = − λ − λ − λ 2 λ Figure 15: The case ρ = − λ 17 λ y 1 λ x 1 λ y 2 λ x 2 λ x 3 λ − 2 λ − 2 λ − 2 λ V λ ( x 1 , x 2 , x 3 | y 1 , y 2 ) λ y 1 λ x 1 λ x 2 λ − 2 λ − 2 λ V λ ( x 1 , x 2 | y 1 ) λ x 1 λ − 2 λ V λ ( x 1 | ∅ ) Figure 16: Kernels of raising operators 2.2 Raising op erator and reflection symmetry The raising op erator V n ( λ ) for B C T o da c hain is an integral operator acting on functions φ ( x n − 1 )  V n ( λ ) φ  ( x n ) = Z R n − 1 d y n − 1 V λ ( x n | y n − 1 ) φ ( y n − 1 ) (2.7) with the k ernel V λ ( x n | y n − 1 ) = (2 β ) i λ Γ( g − i λ ) Z R n d z n (1 + β e − z 1 ) − i λ − g (1 − β e − z 1 ) − i λ + g − 1 θ ( z 1 − ln β ) × exp  i λ  x n + y n − 1 − 2 z n  − n − 1 X j =1 ( e z j − x j + e z j − y j + e x j − z j +1 + e y j − z j +1 ) − e z n − x n  . (2.8) It is easy to understand the structure of this integral by lo oking at the corresp onding diagrams in Figure 16 . F or the parameter of the raising op erator we alwa ys assume λ ∈ R , and the kernel is clearly absolutely con vergen t in this case (since g > 0). Recall the space of con tinuous p olynomially b ounded functions P n =  φ ∈ C ( R n ) : | φ ( x n ) | ≤ P ( | x 1 | , . . . , | x n | ) , P — p olynomial  . (2.9) By Prop osition B.2 the raising op erator resp ects this space V n ( λ ) : P n − 1 → P n , λ ∈ R . (2.10) In this section w e prov e the reflection symmetry . Prop osition 2.1. F or λ ∈ R the r elation V n ( λ ) = V n ( − λ ) (2.11) holds on the sp ac e P n − 1 . 18 Pr o of. Since b oth sides are well defined on P n − 1 , it is sufficient to establish the equalit y of k ernels V λ ( x n | y n − 1 ) = V − λ ( x n | y n − 1 ) . (2.12) This can b e done using three types of diagram transformations, as shown in Figure 17 . The first step is to use r e duc e d cross relation from Figure 6 . As a result, the dashed line app ears at the b ottom, and spectral parameters b elo w it change signs. After that we apply ful l cross relation from Figure 6 in order to mov e this dashed line to the top rhombus. Again, sp ectral parameters on its wa y change signs. Finally , w e use identit y depicted in Figure 15 . This remo ves dashed line from the picture, while changing the signs of sp ectral parameters ab o v e it. The resulting diagram represen ts k ernel of the op erator from the righ t hand side of identit y ( 2.11 ). Although Figure 17 corresp onds to the case n = 3, generalization to arbitrary n is simple: with many rhom buses in the middle one just uses cross relation many times to mov e dashed line from b ottom to top. λ λ λ λ λ λ − 2 λ − 2 λ = − 2 λ λ λ λ λ λ − λ 2 λ − 2 λ − 2 λ = 2 λ λ λ λ − λ − λ − λ 2 λ − 2 λ 2 λ 2 λ − λ − λ − λ − λ − λ − λ 2 λ 2 λ = 2 λ Figure 17: Reflection symmetry of raising op erator 19 λ y 1 λ x 1 λ y 2 λ x 2 λ − 2 λ − 2 λ − 2 λ Q λ ( x 1 , x 2 | y 1 , y 2 ) − λ y 1 − λ x 1 − λ y 2 − λ x 2 − λ 2 λ 2 λ Q ′ λ ( x 1 , x 2 | y 1 , y 2 ) λ y 1 λ x 1 λ − 2 λ − 2 λ Q λ ( x 1 | y 1 ) Figure 18: Kernels of Baxter operators 2.3 Baxter op erator and its reduction Baxter op erator Q n ( λ ) is an in tegral op erator acting on functions φ ( x n )  Q n ( λ ) φ  ( x n ) = Z R n d y n Q λ ( x n | y n ) φ ( y n ) , (2.13) where the k ernel Q λ ( x n | y n ) = (2 β ) i λ Γ( g − i λ ) Z R n +1 d z n +1 (1 + β e − z 1 ) − i λ − g (1 − β e − z 1 ) − i λ + g − 1 θ ( z 1 − ln β ) × exp  i λ  x n + y n − 2 z n +1  − n X j =1 ( e z j − x j + e z j − y j + e x j − z j +1 + e y j − z j +1 )  . (2.14) Let us define the closely related r e duc e d Baxter op erator Q ′ n ( λ ) as an in tegral op erator  Q ′ n ( λ ) φ  ( x n ) = Z R n d y n Q ′ λ ( x n | y n ) φ ( y n ) , (2.15) whose k ernel is given by a slightly smaller in tegral Q ′ λ ( x n | y n ) = (2 β ) − i λ Γ( g + i λ ) Z R n d z n  1 + β e − z 1  i λ − g  1 − β e − z 1  i λ + g − 1 θ ( z 1 − ln β ) (2.16) × exp  i λ  2 z n − x n − y n  − n − 1 X j =1 ( e z j − x j + e z j − y j + e x j − z j +1 + e y j − z j +1 ) − e z n − x n − e z n − y n  . The examples of these k ernels are depicted in Figure 18 . Notice that sp ectral parameters in the k ernels of these op erators ha ve opp osite signs. 20 By Proposition B.2 the abov e Baxter operators act in v arian tly on the space of con tinuous p olynomially b ounded functions ( 2.9 ) Q n ( λ ) : P n → P n , Im λ ∈ ( − g , 0) , (2.17) Q ′ n ( λ ) : P n → P n , Im λ < 0 . (2.18) With the ab o v e assumptions on parameters their kernels are absolutely conv ergen t. Prop osition 2.2. F or Im λ ∈ ( − g , 0) the r elation Q n ( λ ) = Γ(2i λ ) Q ′ n ( λ ) (2.19) holds on P n . Pr o of. Since b oth sides are w ell defined on P n , it is sufficient to sho w the equalit y of kernels, whic h can b e done with the help of diagrams, see Figure 19 . On the first step one applies chain relation from Figure 5 to obtain dashed line at the b ottom, while the rest of transformations coincide with the ones in Figure 17 . In the case n = 0, in accordance with ( 2.19 ) and the ab o v e definitions, we denote Q ′ 0 ( λ ) = (2 β ) − i λ Γ( g + i λ ) Id , Q 0 ( λ ) = (2 β ) − i λ Γ(2i λ ) Γ( g + i λ ) Id . (2.20) λ λ λ λ λ − 2 λ − 2 λ = Γ(2i λ ) − 2 λ λ λ λ λ λ 2 λ − 2 λ − 2 λ = Γ(2i λ ) λ λ λ − λ − λ 2 λ − 2 λ 2 λ − λ − λ − λ − λ − λ 2 λ 2 λ = Γ(2i λ ) Figure 19: Reduction of Baxter operator 21 2.4 Comm utativit y and exchange relations Prop osition 2.3. The r elations Q n ( λ ) Q n ( ρ ) = Q n ( ρ ) Q n ( λ ) , Im λ, Im ρ ∈ ( − g , 0) , (2.21) Q ′ n ( λ ) Q ′ n ( ρ ) = Q ′ n ( ρ ) Q ′ n ( λ ) , Im λ, Im ρ < 0 , (2.22) Q n ( λ ) V n ( ρ ) = Γ(i λ ± i ρ ) V n ( ρ ) Q n − 1 ( λ ) , Im λ ∈ ( − g , 0) , ρ ∈ R , (2.23) Q ′ n ( λ ) V n ( ρ ) = Γ(i λ ± i ρ ) V n ( ρ ) Q ′ n − 1 ( λ ) , Im λ < 0 , ρ ∈ R , (2.24) V n +1 ( λ ) V n ( ρ ) = V n +1 ( ρ ) V n ( λ ) , λ, ρ ∈ R , (2.25) hold on the sp ac es P n and P n − 1 c orr esp ondingly. Pr o of. By Prop osition 2.2 , Q n ( λ ) = Γ(2i λ ) Q ′ n ( λ ), so it is enough to pro ve the relations with reduced Baxter op erators Q ′ n ( λ ). Comm utativit y ( 2.22 ) . In App endix B.4 w e sho w that it is sufficient to establish the equalit y of the corresp onding kernels Z R n d y n Q ′ λ ( x n | y n ) Q ′ ρ ( y n | z n ) = Z R n d y n Q ′ ρ ( x n | y n ) Q ′ λ ( y n | z n ) . (2.26) This can b e done using transformations of diagrams, as shown in Figure 20 . Let us postp one the conv ergence questions to the end of the pro of. As b efore, the pictures corresp ond to the case n = 3, ho wev er, generalization to arbitrary n is straigh tforward. The first diagram in Figure 20 depicts kernel of the pro duct Q ′ 3 ( λ ) Q ′ 3 ( ρ ). Commutativit y is equiv alent to the symmetry of this kernel with resp ect to λ, ρ . The first step is to use chain relation (Figure 5 ) for the b old v ertex at the b ottom to obtain the second diagram with dashed line. F or brevity , w e omit co efficient Γ(i λ + i ρ ) app earing in this transformation, since it is symmetric in λ, ρ . Second, we apply cross relation (Figure 6 ) tw o times in order to mov e dashed line to the top. Notice that some sp ectral parameters b elow change. Next, we use iden tity from Figure 9 that interc hanges parameters λ, ρ in double lines. Due to this identit y tw o more dashed lines app ear. These new dashed lines then can b e brought to the b ottom of diagram with the help of cross relations (Figure 6 ). Finally , dashed lines at the b ottom of the fifth diagram can b e remov ed using reduced cross relations (Figure 6 ). As a result we arrive at the sixth diagram, whic h differs from the thir d one only b y interc hange of λ and ρ . Therefore, the initial integral is symmetric in λ, ρ , which prov es comm utativity of Baxter op erators. It is left to argue that eac h diagram on the w ay corresp onds to the absolutely conv ergent in tegral. Consider any diagram and denote the corresp onding in tegral as D λ,ρ ( x n | y n ). F rom definitions of the lines (Figures 1 , 2 ) w e hav e   D λ,ρ ( x n | y n )   ≤ C ( λ, ρ ) D i Im λ, i Im ρ ( x n | y n ) . (2.27) Hence, it is enough to consider the case λ, ρ ∈ i R . In App endix B.3 we pro v e that the kernel of the Baxter op erators pro duct ( 2.26 ), whic h corresponds to the first diagram, is con vergen t. Since the 22 next diagram differs only b y one transformation, the corresp onding integral is con vergen t at least in one order (suc h that the transformed integral is tak en in the first place). By F ubini–T onelli theorem it is therefore conv ergent in an y order. The third diagram differs by t w o transformations, and for both of them w e can inductiv ely apply the same argumen ts. This induction can be con tinued until the last diagram, so that all diagrams represent con vergen t integrals. Relation ( 2.24 ) . Again, as argued in Section B.4 , it is sufficient to establish the equalit y of the corresp onding kernels Z R n d y n Q ′ λ ( x n | y n ) V ρ ( y n | z n − 1 ) = Γ(i λ ± i ρ ) Z R n − 1 d y n − 1 V ρ ( x n | y n − 1 ) Q ′ λ ( y n − 1 | z n − 1 ) . (2.28) With the help of diagrams it can be done almost in the same w ay , as for comm utativity of Baxter op erators. The first diagram from Figure 21 depicts the kernel of operator Q ′ n ( λ ) V n ( − ρ ). Here w e reflect the parameter of raising operator ρ to ease comparison with Figure 20 . Namely , notice that the first diagrams of Figures 20 and 21 coincide up to one line at the b ottom. T o arriv e at the second diagram of Figure 21 one rep eats the same four steps, as in Figure 20 . The rest of transformations are sligh tly different. Passing to the third diagram we use reduced cross relation (Figure 6 ) for the b ottom v ertex from the left and c hain relation (Figure 5 ) for the b ottom v ertex from the righ t. The resulting diagram contains only one dashed line, whic h we mo ve to the b ottom using cross relation (Figure 6 ). Finally , to arrive at the last diagram, which depicts k ernel of op erator V n ( − ρ ) Q ′ n − 1 ( λ ), w e again use reduced cross relation (Figure 6 ). Note that gamma functions, whic h en ter the righ t hand side of identit y ( 2.24 ), app ear during the first tw o transitions of Figure 21 . This finishes the pro of of desired relation. By the same argumen ts, as b efore, all diagrams corresp ond to absolutely conv ergent integrals. Relation ( 2.25 ) . The pro of is analogous to the previous ones. Remark 2. Since the first diagrams of Figures 20 and 21 coincide up to one line at the b ottom, the latter can b e obtained as the limit of the former Kernel  Q ′ n ( λ ) V n ( − ρ )  ( x n | y n − 1 ) = lim y n →∞ e i ρy n Kernel  Q ′ n ( λ ) Q ′ n ( ρ )  ( x n | y n ) . (2.29) Th us, once the comm utativity of Baxter op erators is established, one can deduce the relation b et w een Baxter and raising operators ( 2.24 ) from it by taking the ab o v e limit. This approach has b een applied for Calogero–Ruijsenaars t yp es of mo dels [ BDKK , BCDK ], which lac k lo cal diagrams transformations. 23 − λ − λ − λ − λ 2 λ 2 λ 2 λ − ρ − ρ − ρ − ρ 2 ρ 2 ρ 2 ρ − λ − ρ − λ − ρ − λ − ρ − λ − λ − λ − λ 2 λ 2 λ 2 λ − ρ − ρ − ρ − ρ 2 ρ 2 ρ 2 ρ − λ − ρ − λ − ρ λ + ρ − λ − λ − λ − λ 2 λ λ − ρ λ − ρ − ρ − ρ − ρ − ρ 2 ρ ρ − λ ρ − λ λ + ρ λ + ρ λ + ρ − ρ − ρ − λ − λ 2 ρ λ − ρ λ − ρ λ − ρ − λ − λ − ρ − ρ 2 λ ρ − λ ρ − λ λ + ρ λ + ρ ρ − λ λ + ρ − ρ − ρ − ρ − λ 2 ρ ρ − λ λ − ρ λ − ρ − λ − λ − λ − ρ 2 λ λ − ρ ρ − λ ρ − λ λ + ρ λ + ρ λ + ρ − ρ − ρ − ρ − ρ 2 ρ ρ − λ ρ − λ − λ − λ − λ − λ 2 λ λ − ρ λ − ρ λ + ρ λ + ρ λ + ρ Figure 20: Commutativit y of Baxter op erators 24 − λ − λ − λ − λ 2 λ 2 λ 2 λ − ρ − ρ − ρ 2 ρ 2 ρ 2 ρ − λ − ρ − λ − ρ − λ − ρ − ρ − ρ − ρ − λ 2 ρ ρ − λ λ − ρ λ − ρ − λ − λ − λ 2 λ λ − ρ ρ − λ ρ − λ λ + ρ λ + ρ λ + ρ − ρ − ρ − ρ − ρ 2 ρ ρ − λ ρ − λ − λ − λ − λ 2 λ λ − ρ λ + ρ λ + ρ λ + ρ − ρ − ρ − ρ − ρ 2 ρ 2 ρ ρ − λ − λ − λ − λ 2 λ 2 λ − λ − ρ λ + ρ λ + ρ − ρ − ρ − ρ − ρ 2 ρ 2 ρ 2 ρ − λ − λ − λ 2 λ 2 λ − λ − ρ − λ − ρ see Fig. 20 Figure 21: Lo cal relation b et ween Baxter and raising op erators 25 λ 3 λ + 23 x 1 λ 3 λ + 23 x 2 λ 3 x 3 λ 3 λ 2 λ + 12 λ 1 − 2 λ 3 − 2 λ 3 − 2 λ 3 − 2 λ 2 − 2 λ 2 − 2 λ 1 Ψ λ 1 ,λ 2 ,λ 3 ( x 1 , x 2 , x 3 ) x 1 λ 3 λ − 23 x 2 λ 3 λ − 23 x 3 λ 3 λ − 12 Φ λ 1 ,λ 2 ,λ 3 ( x 1 , x 2 , x 3 ) Figure 22: W av e functions of GL and B C systems ( λ ± j k = λ j ± λ k ) 2.5 Prop erties of wa ve functions The w av e functions of B C T o da chain are giv en by iterative integrals Ψ λ n ( x n ) = V n ( λ n ) · · · V 1 ( λ 1 ) · 1 , (2.30) where w e assume λ n ∈ R n . This m ultiple integral is easy to visualize using diagrams, see Fig- ure 22 , where for comparison w e also depict wa ve functions of GL T o da c hain defined by ( 1.20 ). Theorem 2.1. L et σ ∈ S n , ε n ∈ { 1 , − 1 } n . Then Ψ λ 1 ,...,λ n ( x n ) = Ψ ε 1 λ σ (1) ,...,ε n λ σ ( n ) ( x n ) . (2.31) Pr o of. By Prop osition B.2 , the raising op erators resp ect the spaces of p olynomially b ounded con tinuous functions V k ( λ k ) : P k − 1 → P k , λ k ∈ R , (2.32) whic h implies the con vergence of the integral ( 2.30 ). Besides, b y Prop ositions 2.1 , 2.3 , the raising op erators satisfy the identities V k ( λ ) = V k ( − λ ) , V k +1 ( λ ) V k ( ρ ) = V k +1 ( ρ ) V k ( λ ) (2.33) that hold on P k − 1 . This leads to the claim. Corollary 2.1. L et λ n ∈ R n . Then Ψ λ n ( x n ) ∈ R . Pr o of. By definition ( 2.8 ), for λ ∈ R V λ ( x n | y n − 1 ) = V − λ ( x n | y n − 1 ) , (2.34) 26 and consequen tly , for λ n ∈ R n Ψ λ n ( x n ) = Ψ − λ n ( x n ) . (2.35) Hence, b y Theorem 2.1 the wa v e function is real. Theorem 2.2. L et λ n ∈ R n . Then Q n ( λ ) Ψ λ n ( x n ) = (2 β ) − i λ Γ(2i λ ) Γ( g + i λ ) n Y j =1 Γ(i λ ± i λ j ) Ψ λ n ( x n ) , Im λ ∈ ( − g , 0) , (2.36) Q ′ n ( λ ) Ψ λ n ( x n ) = (2 β ) − i λ Γ( g + i λ ) n Y j =1 Γ(i λ ± i λ j ) Ψ λ n ( x n ) , Im λ < 0 . (2.37) Pr o of. By Prop osition 2.3 , Baxter operator and its reduced version satisfy the relations (which hold on P n − 1 ) Q n ( λ ) V n ( ρ ) = Γ(i λ ± i ρ ) V n ( ρ ) Q n − 1 ( λ ) , Im λ ∈ ( − g , 0) , ρ ∈ R , (2.38) Q ′ n ( λ ) V n ( ρ ) = Γ(i λ ± i ρ ) V n ( ρ ) Q ′ n − 1 ( λ ) , Im λ < 0 , ρ ∈ R , (2.39) where in particular Q 0 ( λ ) = (2 β ) − i λ Γ(2i λ ) Γ( g + i λ ) Id , Q ′ 0 ( λ ) = (2 β ) − i λ Γ( g + i λ ) Id . (2.40) Th us, from iterative representation ( 2.30 ) we obtain the claimed form ulas. 2.6 Lo cal relation b etw een GL and B C op erators F or any integral op erator K [ K φ ]( x n ) = Z R m d y m K ( x n | y m ) φ ( y m ) (2.41) denote its transp ose K t b y the formula [ K t φ ]( x m ) = Z R n d y n K ( y n | x m ) φ ( y n ) . (2.42) The following identit y is the key ingredient in the calculation of the scalar pro duct b et w een GL and B C wa v e functions, p erformed in the next section. Prop osition 2.4. L et n ≥ 2 , Im λ < 0 and ρ ∈ R . Then the r elation Q t n − 1 ( λ ) Λ t n ( − λ ) exp( − β e − x 1 ) V n ( ρ ) = Γ(i λ ± i ρ ) Q t n − 1 ( ρ ) Q t n − 1 ( − ρ ) exp( − β e − x 1 ) Q ′ n − 1 ( λ ) (2.43) holds as the e quality b etwe en c orr esp onding kernels. 27 Pr o of. The proof of the iden tity ( 2.43 ) in terms of diagrams is shown in Figures 23 , 24 . In Section B.3 w e pro ve that the kernel of the left hand side ( 2.43 ), whic h corresp onds to the first diagram, is absolutely conv ergen t. By the same argumen ts, as in the proof of Prop osition 2.3 , all of the rest diagrams also corresp ond to absolutely conv ergent integrals. As b efore, the pictures are for the particular case n = 3, while generalization to arbitrary n is straightforw ard. In all diagrams we indicate coordinate ln β , which comes from the function exp( − β e − x 1 ) in ( 2.43 ). The first step is to use c hain relation (Figure 5 ) for the vertex at the b ottom and obtain one dashed line. Then one mo ves this dashed line to the top using cross relations (Figure 6 ). After that w e rep eat the same pro cedure for the vertex at the bottom from the right. Doing so we arrive at the fifth diagram with tw o dashed lines at the top. The next step is to use iden tit y from Figure 14 , whic h remov es one dashed line and changes some sp ectral parameters. In particular, the parameter of double line is c hanged ρ → λ . Notice that for this iden tity it is crucial that the top dashed line connects double line with the co ordinate ln β . On the next step, sho wn in Figure 24 , w e use the iden tity pictured in Figure 25 . It represen ts comm utativity of GL T o da Q -op erators (see ( 1.27 ), ( 1.40 )). This step enables us to again use cross relation (Figure 6 ) to mo ve dashed line to the b ottom. P assing to the last diagram we use reduced cross relation (Figure 6 ), so that the dashed line disapp ears, and we arriv e at the k ernel of op erator from the right hand side of iden tity ( 2.43 ). Collecting factors appearing in all abov e transformations w e obtain the co efficien t Γ(i λ ± i ρ ). Remark 3. With the help of [ BDK , Corollary 2] together with b ounds from App endix B one can sho w that the identit y ( 2.43 ) holds on P n − 1 . 28 ln β − λ − λ 2 λ 2 λ ρ ρ ρ − 2 ρ − 2 ρ − 2 ρ ρ − λ ρ − λ ρ − λ ln β − λ − λ 2 λ 2 λ ρ ρ ρ − 2 ρ − 2 ρ − 2 ρ ρ − λ ρ − λ λ − ρ ln β − λ − λ λ + ρ λ + ρ ρ ρ ρ − 2 ρ − λ − ρ − λ − ρ λ − ρ λ − ρ λ − ρ ln β − λ − λ λ + ρ λ + ρ ρ ρ ρ − 2 ρ − λ − ρ λ + ρ λ − ρ λ − ρ λ − ρ ln β − λ − λ λ + ρ λ + ρ ρ ρ − λ − 2 ρ λ + ρ λ + ρ λ − ρ − 2 ρ λ − ρ ln β − λ − λ λ + ρ λ + ρ − λ − λ − λ 2 λ λ + ρ − 2 ρ − 2 ρ λ − ρ Figure 23: Pro of of identit y ( 2.43 ) (contin ues in Figure 24 ) 29 ln β − λ − λ λ + ρ λ + ρ − λ − λ − λ 2 λ λ + ρ − 2 ρ − 2 ρ λ − ρ ln β ρ ρ − λ − ρ − λ − ρ − λ − λ − λ 2 λ λ + ρ λ − ρ λ − ρ λ − ρ ln β ρ ρ − 2 ρ − λ − ρ − λ − λ − λ 2 λ 2 λ ρ − λ λ − ρ λ − ρ ln β ρ ρ − 2 ρ − 2 ρ − λ − λ − λ 2 λ 2 λ ρ − λ ρ − λ see Fig. 25 Figure 24: Pro of of identit y ( 2.43 ) (b egins in Figure 23 ) − ρ − ρ = − λ − λ λ + ρ λ + ρ λ λ ρ ρ − λ − ρ − λ − ρ Figure 25: Commutativit y of GL T o da Q -op erators 30 2.7 Scalar pro duct b etw een GL and B C wa ve functions The follo wing prop osition is needed to deriv e Mellin–Barnes representation of the B C wa v e function. Prop osition 2.5. L et λ n ∈ R n , ρ n ∈ C n such that Im ρ 1 = . . . = Im ρ n < 0 . Then Z R n d x n Φ − ρ n ( x n ) Ψ λ n ( x n ) exp( − β e − x 1 ) = (2 β ) − i ρ n n Q j,k =1 Γ(i ρ j ± i λ k ) Q 1 ≤ j 0 Ψ λ n ( x n ) = e β e − x 1 Z ( R − i ε ) n d γ n ˆ µ ( γ n ) K 1 ( λ n , γ n ) Φ γ n ( x n ) . (2.66) Pr o of. The pro of relies on Prop osition 2.5 and inv ersion form ula for GL T o da wa ve function. Namely , it is kno wn [ W1 ] that smo oth and sufficien tly fast deca ying functions φ ( x n ), whic h b elong to the so called Whittaker Schwartz sp ac e , can b e expanded in terms of GL T o da wa v e functions φ ( x n ) = Z R n d γ n ˆ µ ( γ n ) Φ γ n ( x n ) Z R n d y n Φ γ n ( y n ) φ ( y n ) . (2.67) Equiv alently , GL w av e functions satisfy completeness relation ( 1.45 ). The definition of Whit- tak er Sch w artz space is given in Section B.5 . The form ula ( 2.66 ) can b e written in the same spirit. Namely , using the relation ( 1.23 ) Φ γ n − i ε e n ( x n ) = e ε x n Φ γ n ( x n ) , e n = (1 , . . . , 1) , (2.68) and shifting in tegration v ariables γ j → γ j − i ε w e arrive at e − ε x n − β e − x 1 Ψ λ n ( x n ) = Z R n d γ n ˆ µ ( γ n ) K 1 ( λ n , γ n − i ε e n ) Φ γ n ( x n ) . (2.69) By Corollary B.2 , the function e − ε x n − β e − x 1 Ψ λ n ( x n ) belongs to Whittaker Sc hw artz space, and hence w e are allow ed to apply in version formula ( 2.67 ) to it. It is left to recall Prop osition 2.5 , whic h states that the scalar pro duct of this function with GL T o da w av e function coincides with the kernel ( 2.65 ). Namely , for γ n ∈ R n w e hav e Φ γ n ( x n ) = Φ − γ n ( x n ) (2.70) see ( 1.17 ), so that Z R n d x n Φ γ n ( x n ) e − ε x n − β e − x 1 Ψ λ n ( x n ) = Z R n d x n Φ − γ n +i ε e n ( x n ) e − β e − x 1 Ψ λ n ( x n ) = K 1 ( λ n , γ n − i ε e n ) . (2.71) 34 2.9 Orthogonalit y The Hamiltonians of B C T o da c hain are formally self-adjoin t with respect to the standard scalar pro duct ⟨ ψ | φ ⟩ = Z R n d x n ψ ( x n ) φ ( x n ) . (2.72) In this section we presen t heuristic calculation of the scalar pro duct b et ween B C wa v e functions ⟨ Ψ λ n | Ψ ρ n ⟩ = 1 ˆ µ B C ( λ n ) δ sym ( λ n , ρ n ) , λ n , ρ n ∈ R n , (2.73) where w e denote the measure ˆ µ B C ( λ n ) = 1 n ! (4 π ) n Y 1 ≤ j 0. It is absolutely con vergen t for | Im λ j | < ε and analytically con tinues to λ n ∈ C n b y shifting con tours (i.e. increasing ε ), see Prop osition C.2 and Corollary C.2 . Thus, the action of dual Hamiltonians ( 3.26 ) on these functions is w ell defined, and with the help of Prop osition 3.1 w e can prov e the follo wing statement. Theorem 3.1. The wave function Ψ λ n ( x n ) satisfies van Diejen–Emsiz e quations ˆ H s Ψ λ n ( x n ) = e x n − s +1 + ... + x n Ψ λ n ( x n ) (3.43) for s = 1 , . . . , n . 45 Pr o of. By Prop osition C.2 , the integral ( 3.42 ) con verges uniformly in λ j from compact subsets of the strips | Im λ j | < ε . Hence, taking ε > 1 we can in terchange difference op erators ˆ H s with in tegration in ( 3.43 ) and then use Prop osition 3.1 ˆ H s ( λ n ) Ψ λ n ( x n ) = e β e − x 1 Z ( R − i ε ) n d γ n ˆ µ ( γ n ) h ˆ H s ( λ n ) K 1 ( λ n , γ n ) i Φ γ n ( x n ) = e β e − x 1 Z ( R − i ε ) n d γ n ˆ µ ( γ n ) h ˆ H s ( − γ n ) K 1 ( λ n , γ n ) i Φ γ n ( x n ) . (3.44) Next shift in tegration v ariables γ j = ρ j − i ε , so that ρ j ∈ R and Φ γ n ( x n ) = e ε x n Φ ρ n ( x n ) = e ε x n Φ − ρ n ( x n ) , (3.45) see ( 1.17 ) and ( 1.20 ). As a result, ˆ H s ( λ n ) Ψ λ n ( x n ) = e β e − x 1 + ε x n Z R n d ρ n ˆ µ ( ρ n ) h ˆ H s ( − ρ n ) K 1 ( λ n , ρ n − i ε e n ) i Φ − ρ n ( x n ) . (3.46) No w recall that GL dual Hamiltonians are symmetric with resp ect to the scalar product ( 3.3 ) with the measure ˆ µ ( ρ n ) = ˆ µ ( − ρ n ) ˆ H s ( λ n ) Ψ λ n ( x n ) = e β e − x 1 + ε x n Z R n d ρ n ˆ µ ( ρ n ) K 1 ( λ n , ρ n − i ε e n ) h ˆ H s ( − ρ n ) Φ − ρ n ( x n ) i . (3.47) It is left to use the fact that Φ − ρ n ( x n ) diagonalize Hamiltonians ˆ H s ( 3.19 ). In a similar w ay Prop osition 3.1 implies that the function ˜ Ψ λ n ( x n ) = e − β e − x 1 Z ( R − i0) n d γ n µ ( γ n ) K 2 ( λ n , γ n )Φ γ n ( x n ) (3.48) also solv es v an Diejen–Emsiz equations ˆ H s ˜ Ψ λ n ( x n ) = e x n − s +1 + ... + x n Ψ λ n ( x n ) . By Prop osition C.3 the in tegral ( 3.48 ) is absolutely conv ergen t for | Im λ j | < ε < g , so that for the last prop ert y w e in addition assume g > 1. As explained in Section 1.3 , in the case n = 1 the Mellin–Barnes in tegrals ( 3.42 ), ( 3.48 ) represen t tw o known expressions for Whittaker function, see Examples 2 , 3 . Let us pro ve that they also coincide in the general case. Prop osition 3.2. F or λ n ∈ R n and ε ∈ (0 , g ) Ψ λ n ( x n ) = e − β e − x 1 Z ( R − i ε ) n d γ n ˆ µ ( γ n ) K 2 ( λ n , γ n ) Φ γ n ( x n ) . (3.49) 46 Pr o of. W e hav e to pro ve the equality e β e − x 1 Z ( R − i ε ) n d γ n ˆ µ ( γ n ) n Q j,k =1 Γ(i γ j ± i λ k ) Q 1 ≤ j 0 and w e denote Q ( λ n , t n ) = (2 β ) − i t n n Q a,i =1 Γ (i( ± λ i + t a )) Q a  = b Γ (i( t a − t b )) Q a 0 (3.110) w e hav e a b ound | Φ λ n ( x n ) − Φ as λ n ( x n ) | < C ( λ n ) exp( − ε/ 2) (3.111) 55 for an y λ n ∈ R n , suc h that λ i  = λ j . The eigenfunction of GL T o da Hamiltonians analytical in the region x 1 < . . . < x n with plane wa ve asymptotic e i P k λ k x k is usually called the Harish-Chandra function φ H C λ n ( x n ) [ HC ]. It can b e represented in a form of con vergen t series ov er v ariables e − ( x k − x k +1 ) [ DE1 , DE2 , O ], and after the symmetrization ov er sp ectral parameters it gives the eigenfunction symmetric and analytical o ver sp ectral parameters. Th us, φ λ n ( x n ) = Y i 0 (3.122) w e hav e the b ound | Ψ λ n ( x n ) − Ψ as λ n ( x n ) | < C ( λ n ) exp( − ε/ 2) (3.123) for any λ n ∈ R n , such that λ i  = ± λ j and λ j  = 0. This is precisely the asymptotics of the hype- ro ctahedral Whittaker function [ DE2 , (3.5b)] (to compare set 2 β = 1). W e th us conclude that Ψ λ n ( x n ) coincides with hypero ctahedral Whittaker function due to uniqueness of the latter, and the relation ( 3.120 ) is its Harish-Chandra decomp osition. As shown in [ DE2 ], h yp ero c- tahedral Whittak er function is entire in sp ectral parameters, whic h is in accordance with our Corollary C.2 . 57 3.2.5 Completeness As w e already discussed, the wa v e functions of GL T o da chain satisfy completeness relation ( 3.22 ) Z R n d λ n ˆ µ ( λ n ) Φ λ n ( x n ) Φ λ n ( y n ) = δ ( x 1 − y 1 ) · · · δ ( x n − y n ) , (3.124) where x n , y n ∈ R n and the measure is giv en by ( 3.4 ). In this section w e present heuristic deriv ation of the completeness relation for B C w av e functions Z R n d λ n ˆ µ B C ( λ n ) Ψ λ n ( x n ) Ψ λ n ( y n ) = δ ( x 1 − y 1 ) · · · δ ( x n − y n ) , (3.125) where the corresp onding measure is determined by orthogonalit y relation ˆ µ B C ( λ n ) = 1 n ! (4 π ) n Y 1 ≤ j ln β , y > ln β and also for conv ergence Re(i ρ ) < Re(i λ ) < g . T o pro ve this symmetry it is con venien t to pass to the exp onen tial v ariables X = e x > β , Y = e y > β , Z = e z . (A.3) Then the function can b e rewritten in the form F = X i( λ − ρ ) Y − 2i ρ (1 + β / Y ) − i ρ − g (1 − β / Y ) − i ρ + g − 1 × Z X β d Z Z − 2i λ − 1 (1 + β / Z ) − i λ − g (1 − β / Z ) − i λ + g − 1 ( Z + Y ) i( λ + ρ ) (1 − Z/X ) i( λ − ρ ) − 1 . (A.4) Next to remov e dep endence on external parameters from the b ounds of integral w e c hange in tegration v ariable w = X − Z Z − β . (A.5) As a result, w e arrive at the expression F = X Y ( X − β ) − i ρ + g − 1 ( Y − β ) − i ρ + g − 1 ( Y + β ) − i ρ − g × Z ∞ 0 dw w i( λ − ρ ) − 1 ( X + β + 2 β w ) − i λ − g  X + Y + ( Y + β ) w  i( λ + ρ ) . (A.6) It is left to rescale in tegration v ariable w = ( X + β ) s (A.7) to obtain the form ula F = X Y ( X − β ) − i ρ + g − 1 ( Y − β ) − i ρ + g − 1 ( X + β ) − i ρ − g ( Y + β ) − i ρ − g × Z ∞ 0 ds s i( λ − ρ ) − 1 (1 + 2 β s ) − i λ − g  X + Y + ( X + β )( Y + β ) s  i( λ + ρ ) , (A.8) whic h is clearly symmetric in X , Y . A.2 Pro of of reduced flip relation The reduced flip relation pictured in Figure 12 represen ts the following identit y e i( λ − ρ ) x Z x ln β dz e − 2i λz (1 + β e − z ) − i λ − g (1 − β e − z ) − i λ + g − 1 × ( e z + β ) i( λ + ρ ) (1 − e z − x ) i( λ − ρ ) − 1 = (2 β ) i( ρ − λ ) Γ(i λ − i ρ ) Γ( g − i λ ) Γ( g − i ρ ) × e − 2i ρx (1 + β e − x ) − i ρ − g (1 − β e − x ) − i ρ + g − 1 ( e x + β ) i( λ + ρ ) , (A.9) where w e assume x > ln β . Note also that the integral is absolutely con vergen t under condition Re(i ρ ) < Re(i λ ) < g . 60 T o prov e the ab ov e identit y consider asymptotics of the function F ( x, y ) defined by ( A.2 ) as y → ln β + . F rom its definition we hav e F ( x, y ) = y → ln β + ( y − ln β ) − i ρ + g − 1 2 − i ρ − g β − 2i ρ × e i( λ − ρ ) x Z x ln β dz e − 2i λz (1 + β e − z ) − i λ − g (1 − β e − z ) − i λ + g − 1 × ( e z + β ) i( λ + ρ ) (1 − e z − x ) i( λ − ρ ) − 1 . (A.10) Except the first line, this expression coincides with the left hand side of the iden tity ( A.9 ). Next consider the same limit for the form ula ( A.8 ) (which we rewrite in original v ariables) F ( x, y ) = y → ln β + ( y − ln β ) − i ρ + g − 1 2 − i ρ − g β − 2i ρ × e − 2i ρx (1 + β e − x ) − i ρ − g (1 − β e − x ) − i ρ + g − 1 ( e x + β ) i( λ + ρ ) × Z ∞ 0 ds s i( λ − ρ ) − 1 (1 + 2 β s ) i ρ − g . (A.11) The remaining in tegral after rescaling s → s/ 2 β b ecomes Euler’s beta integral Z ∞ 0 ds s i( λ − ρ ) − 1 (1 + 2 β s ) i ρ − g = (2 β ) i( ρ − λ ) Γ(i λ − i ρ ) Γ( g − i λ ) Γ( g − i ρ ) , (A.12) whic h coincides with the co efficient from the right ( A.9 ). Collecting all together w e arrive at the claimed form ula. One subtle p oin t is that to obtain ( A.10 ) we in terc hanged limit and integration. T o justify it w e use dominated con vergence theorem. Namely , in addition to previous assumptions supp ose Re(i λ ) , Re(i ρ ) < 0, this restriction can be remo ved at the end b y analytic contin uation. Since e y ≥ β , we then can use inequality   ( e z + e y ) i( λ + ρ )   = ( e z + e y ) Re(i λ )+Re(i ρ ) ≤ ( e z + β ) Re(i λ )+Re(i ρ ) (A.13) to b ound the in tegrand in ( A.4 ) b y integrable function, whic h do esn’t dep end on y . Analogous argumen ts should b e applied to the limit of the second form ula ( A.8 ). B Gauss–Giv en tal b ounds Denote b y P n the space of con tinuous p olynomially b ounded functions of n v ariables P n =  φ ∈ C ( R n ) : | φ ( x n ) | ≤ P ( | x 1 | , . . . , | x n | ) , P — p olynomial  . (B.1) In this app endix we sho w that Baxter and raising operators of GL and B C T oda systems act in v arian tly on this space. F or this we need the follo wing three bounds. The last tw o are pro ved in [ BDK ], although here w e state their slightly weak er v ersions, which are sufficient in the presen t context. 61 Lemma B.1. L et m ∈ N 0 , κ > 0 and x ∈ R . Then Z R dy | y | m exp  κ ( x − y ) − e x − y  ≤ P ( | x | ) , (B.2) wher e P ( | x | ) ≡ P ( | x | ; m, κ ) is p olynomial in | x | , and c onver genc e of this inte gr al is uniform in x fr om c omp act subsets of R . Lemma B.2. [ BDK , Cor ol lary 1] L et m ∈ N 0 , κ 1 , κ 2 ≥ 0 and x 1 , x 2 ∈ R . Then Z R dy | y | m exp  κ 1 ( x 1 − y ) + κ 2 ( y − x 2 ) − e x 1 − y − e y − x 2  ≤ P ( | x 1 | , | x 2 | ) , (B.3) wher e P ( | x 1 | , | x 2 | ) ≡ P ( | x 1 | , | x 2 | ; m, κ 1 , κ 2 ) is p olynomial in | x j | , and c onver genc e of this inte gr al is uniform in x 1 , x 2 fr om c omp act subsets of R . Lemma B.3. [ BDK , L emma 3] L et m ∈ N 0 , g > 0 , κ ≥ 0 and x ∈ R . Then Z ∞ 0 dy y m (1 + e − y ) − g (1 − e − y ) g − 1 exp  κ ( y + x ) − e y + x  ≤ P ( | x | ) , (B.4) wher e P ( | x | ) ≡ P ( | x | ; m, g , κ ) is p olynomial in | x | , and c onver genc e of this inte gr al is uniform in x fr om c omp act subsets of R . Pr o of of L emma B.1 . Change v ariable in the in tegral in question to z = x − y Z R dy | y | m e κ ( x − y ) − e x − y = Z R dz | x − z | m e κz − e z ≤ Z R dz ( | x | + | z | ) m e κz − e z . (B.5) Then the desired b ound follows from expanding brac kets, since (recall that κ > 0) Z R dz | z | ℓ e κz − e z < ∞ . (B.6) The ab o v e b ound is clearly uniform in x from compact subsets. B.1 GL system Recall GL T o da raising and Baxter operators  Λ n ( λ ) φ  ( x n ) = Z R n − 1 d y n − 1 exp  i λ  x n − y n − 1  − n − 1 X j =1 ( e x j − y j + e y j − x j +1 )  φ ( y n − 1 ) ,  Q n ( λ ) φ  ( x n ) = Z R n d y n exp  i λ  x n − y n  − n − 1 X j =1 ( e x j − y j + e y j − x j +1 ) − e x n − y n  φ ( y n ) , (B.7) where w e alwa ys assume x n ∈ R n . The following axillary operator  Λ ′ n ( λ ) φ  ( x n ) = e − i λx 1  Λ n ( λ ) φ  ( x n ) (B.8) will b e also useful when we pass to the case of B C system. 62 Prop osition B.1. We have Λ ′ n ( λ ) : P n − 1 → P n , Im λ ≥ 0 , (B.9) Λ n ( λ ) : P n − 1 → P n , λ ∈ R , (B.10) Q n ( λ ) : P n → P n , Im λ < 0 . (B.11) Pr o of. First, notice that the second statemen t ( B.10 ) follows from the first one ( B.9 ) due to definition ( B.8 ). F or the first statement by definition we hav e     Λ ′ n ( λ ) φ  ( x n )    ≤ Z R n − 1 d y n − 1 P ( | y 1 | , . . . , | y n − 1 | ) n − 1 Y j =1 e Im λ ( y j − x j +1 ) − e x j − y j − e y j − x j +1 . (B.12) Expanding polynomial in terms of monomials | y 1 | m 1 · · · | y n − 1 | m n − 1 w e obtain the sum with terms factorised in to one dimensional integrals Z R dy j | y j | m j e Im λ ( y j − x j +1 ) − e x j − y j − e y j − x j +1 , (B.13) where Im λ ≥ 0. By Lemma B.2 these integrals are b ounded by p olynomials P ( | x j | , | x j +1 | ) and con verge uniformly in x j , x j +1 . This gives the needed p olynomial estimate     Λ ′ n ( λ ) φ  ( x n )    ≤ P ( | x 1 | , . . . , | x n | ) , (B.14) while uniformit y ensures that  Λ ′ n ( λ ) φ  ( x n ) is con tinuous in x n . No w consider the third statement ( B.11 ). Due to definition     Q n ( λ ) φ  ( x n )    ≤ Z R n d y n P ( | y 1 | , . . . , | y n | ) e − Im λ ( x n − y n ) − e x n − y n × n − 1 Y j =1 e − Im λ ( x j − y j ) − e x j − y j − e y j − x j +1 . (B.15) Again writing p olynomial in terms of monomials we obtain the sum with terms factorised into in tegrals of tw o t yp es Z R dy j | y j | m j e − Im λ ( x j − y j ) − e x j − y j − e y j − x j +1 ( j = 1 , . . . , n − 1) , (B.16) Z R dy n | y n | m n e − Im λ ( x n − y n ) − e x n − y n , (B.17) where Im λ < 0. All of them are p olynomially bounded and con verge uniformly in x j b y Lemmas B.1 , B.2 . As a b y-pro duct, from the abov e proposition we get the follo wing estimate for GL T o da w a ve function Φ λ n ( x n ) = Λ n ( λ n ) · · · Λ 1 ( λ 1 ) · 1 . (B.18) 63 Corollary B.1. L et λ n ∈ R n . Then Φ λ n ( x n ) ∈ P n , and the c orr esp onding inte gr al is absolutely c onver gent. Pr o of. The fact that w av e function b elongs to P n is clear from ( B.10 ). The absolute con v ergence follo ws from the prop ert y of the raising op erators k ernels   Λ λ k ( x k | y k − 1 )   ≤ Λ 0 ( x k | y k − 1 ) , λ k ∈ R , (B.19) see ( B.7 ). B.2 B C system In tro duce t wo axillary op erators  V n ( λ ) φ  ( x n ) = (2 β ) i λ Γ( g − i λ ) Z R n d y n (1 + β e − y 1 ) − i λ − g (1 − β e − y 1 ) − i λ + g − 1 θ ( y 1 − ln β ) × exp  i λ  x n − y n  − n − 1 X j =1 ( e y j − x j + e x j − y j +1 ) − e y n − x n  φ ( y n ) , (B.20)  W n ( λ ) φ  ( x n ) = (2 β ) i λ Γ( g − i λ ) Z R n +1 d y n +1 (1 + β e − y 1 ) − i λ − g (1 − β e − y 1 ) − i λ + g − 1 θ ( y 1 − ln β ) × exp  i λ  x n − y n +1 − y 1  − n X j =1 ( e y j − x j + e x j − y j +1 )  φ ( y n +1 ) . (B.21) Then B C T o da raising and Baxter op erators from Sections 2.2 and 2.3 can b e written as the pro ducts of GL T o da op erators with axillary ones V n ( λ ) = V n ( λ ) Λ n ( − λ ) , Q n ( λ ) = W n ( λ ) Λ ′ n +1 ( − λ ) , Q ′ n ( λ ) = V n ( − λ ) Q n ( λ ) . (B.22) This splitting is easily visualized in the language of diagrams from Section 2.1 . By Prop osition B.1 GL T o da op erators act within the space of p olynomially b ounded con- tin uous functions, so it is left to establish the same thing ab out axillary op erators. Lemma B.4. We have V n ( λ ) : P n → P n , Im λ ≥ 0 , (B.23) W n ( λ ) : P n +1 → P n , Im λ ∈ ( − g , 0) . (B.24) Before we prov e this lemma let us remark that together with Prop osition B.1 and formu- las ( B.22 ) it giv es the following statement. Prop osition B.2. We have V n ( λ ) : P n − 1 → P n , λ ∈ R , (B.25) Q n ( λ ) : P n → P n , Im λ ∈ ( − g , 0) , (B.26) Q ′ n ( λ ) : P n → P n , Im λ < 0 . (B.27) 64 Pr o of of L emma B.4 . First, consider the op erator V n ( λ ). By definition ( B.20 ) we ha ve     V n ( λ ) φ  ( x n )    ≤ (2 β ) − Im λ | Γ( g − i λ ) | Z R n d y n P ( | y 1 | , . . . , | y n | ) n Y j =2 e Im λ ( y j − x j ) − e x j − 1 − y j − e y j − x j × e Im λ ( y 1 − x 1 ) − e y 1 − x 1 (1 + β e − y 1 ) Im λ − g (1 − β e − y 1 ) Im λ + g − 1 θ ( y 1 − ln β ) . (B.28) Expanding p olynomial in terms of monomials | y 1 | m 1 · · · | y n | m n w e obtain sum with terms fac- torised in to integrals of tw o t yp es Z R dy j | y j | m j e Im λ ( y j − x j ) − e x j − 1 − y j − e y j − x j ( j = 2 , . . . , n ) , (B.29) Z ∞ ln β dy 1 | y 1 | m 1 e Im λ ( y 1 − x 1 ) − e y 1 − x 1 (1 + β e − y 1 ) Im λ − g (1 − β e − y 1 ) Im λ + g − 1 , (B.30) where Im λ ≥ 0. In tegrals from the first line are p olynomially b ounded and conv erge uniformly in x j − 1 , x j due to Lemma B.2 . In the integral from the second line we can use inequalit y (1 + β e − y 1 ) Im λ − g = (1 + β e − y 1 ) 2 Im λ (1 + β e − y 1 ) − Im λ − g ≤ 4 Im λ (1 + β e − y 1 ) − Im λ − g , (B.31) and c hange integration v ariable to z = y 1 − ln β , so that Z ∞ ln β dy 1 | y 1 | m 1 e Im λ ( y 1 − x 1 ) − e y 1 − x 1 (1 + β e − y 1 ) Im λ − g (1 − β e − y 1 ) Im λ + g − 1 ≤ 4 Im λ Z ∞ 0 dz ( | z | + | ln β | ) m 1 e Im λ ( z +ln β − x 1 ) − e z +ln β − x 1 (1 + e − z ) − g ′ (1 − e − z ) g ′ − 1 , (B.32) where w e also denoted g ′ = Im λ + g > 0. Then Lemma B.3 sa ys that the last in tegral is bounded p olynomially and conv erges uniformly in x . Hence, the whole right hand side ( B.28 ) is b ounded b y p olynomial in x n and  V n ( λ ) φ  ( x n ) is con tinuous. No w consider the op erator W n ( λ ). By definition ( B.21 ) we ha ve     W n ( λ ) φ  ( x n )    ≤ (2 β ) − Im λ | Γ( g − i λ ) | Z R n +1 d y n +1 P ( | y 1 | , . . . , | y n +1 | ) × e − Im λ ( x n − y n +1 ) − e x n − y n +1 n Y j =2 e − Im λ ( x j − 1 − y j ) − e x j − 1 − y j − e y j − x j × e 2 Im λ y 1 − e y 1 − x 1 (1 + β e − y 1 ) Im λ − g (1 − β e − y 1 ) Im λ + g − 1 θ ( y 1 − ln β ) . (B.33) Again writing p olynomial in terms of monomials we obtain sum with terms factorised in to in tegrals of three types Z R dy n +1 | y n +1 | m n +1 e − Im λ ( x n − y n +1 ) − e x n − y n +1 , (B.34) Z R dy j | y j | m j e − Im λ ( x j − 1 − y j ) − e x j − 1 − y j − e y j − x j ( j = 2 , . . . , n ) , (B.35) Z ∞ ln β dy 1 | y 1 | m 1 e 2 Im λ y 1 − e y 1 − x 1 (1 + β e − y 1 ) Im λ − g (1 − β e − y 1 ) Im λ + g − 1 , (B.36) 65 where Im λ < 0. The first tw o are p olynomially bounded due to Lemmas B.1 and B.2 corre- sp ondingly . In the last integral we can use inequalit y e 2 Im λ y 1 = β 2 Im λ e 2 Im λ ( y 1 − ln β ) ≤ β 2 Im λ (B.37) to b ound it by the in tegral of the same t yp e, as w e already encountered ( B.30 ). The remain- ing steps are the same, as b efore, so that we obtain p olynomial b ound for the whole expres- sion ( B.33 ). B.3 Kernels of op erator pro ducts Raising and Baxter op erators of B C T oda split into pro ducts of t wo simpler op erators ( B.22 ). Besides, in the pap er we consider lo cal relations b et ween v arious pro ducts of the ab o ve op erators, suc h as Q n ( λ ) Q n ( ρ ) = Q n ( ρ ) Q n ( λ ) , Q n ( λ ) Q n ( ρ ) = Q n ( ρ ) Q n ( λ ) , (B.38) and others. Let us pro ve that the k ernels of all these pro ducts are giv en b y absolutely con v ergent in tegrals. F or example, consider the B C Baxter op erator ( B.22 ) Q n ( λ ) = W n ( λ ) Λ ′ n +1 ( − λ ) , (B.39) with Im λ ∈ ( − g , 0). By Prop osition B.2 the follo wing integral is conv ergen t for φ ∈ P n  Q n ( λ ) φ  ( x n ) = Z R n +1 d y n +1 Z R n d z n W λ ( x n | y n +1 ) Λ ′ − λ ( y n +1 | z n ) φ ( z n ) , (B.40) where in the in tegrand we hav e k ernels of op erators W n ( λ ), Λ ′ n +1 ( − λ ). Moreov er, since   W λ ( x n | y n +1 )   =     Γ( g + Im λ ) Γ( g − i λ )     W i Im λ ( x n | y n +1 ) ,   Λ ′ − λ ( y n +1 | z n )   = Λ ′ − i Im λ ( y n +1 | z n ) (B.41) and | φ | is still in P n , the ab ov e integral is absolutely con vergen t. By F ubini–T onelli theorem this means that w e can interc hange the order of in tegrals and write  Q n ( λ ) φ  ( x n ) = Z R n d z n Q λ ( x n | z n ) φ ( z n ) , (B.42) where Q λ ( x n | z n ) = Z R n +1 d y n +1 W λ ( x n | y n +1 ) Λ ′ − λ ( y n +1 | z n ) . (B.43) No w notice that the kernel Λ ′ − λ ( y n +1 | z n ) = n Y j =1 exp  i λ ( z j − y j +1 ) − e y j − z j − e z j − y j +1  , (B.44) 66 as a function of y n +1 , b elongs to P n +1 . Indeed, it is contin uous and b ounded b y constant   Λ ′ − λ ( y n +1 | z n )   ≤ n Y j =1 exp  − Im λ ( z j − y j +1 ) − e z j − y j +1  ≤ C ( λ ) , (B.45) since Im λ < 0. By Lemma B.4 , the operator W n ( λ ) is well-defined on P n +1 , hence, the in te- gral ( B.43 ) is conv ergen t for Im λ ∈ ( − g , 0). F urthermore, b ecause of ( B.41 ) it is absolutely con vergen t. The same arguments can b e applied to all other kernels of op erator pro ducts app earing in the pap er. F or example, the kernel of Q -op erators’ pro duct Q n ( λ ) Q n ( ρ ) = W n ( λ ) Λ ′ n +1 ( − λ ) W n ( ρ ) Λ ′ n +1 ( − ρ ) (B.46) is given by conv ergent integral, since again the kernel of the last op erator Λ ′ − ρ ( y n +1 | z n ) b elongs to P n +1 (as a function of y n +1 ). Let us also consider the op erator pro duct from Section 2.6 Q t n − 1 ( λ ) Λ t n ( − λ ) exp( − β e − x 1 ) V n ( ρ ) = Q t n − 1 ( λ ) Λ t n ( − λ ) exp( − β e − x 1 ) V n ( ρ ) Λ n ( − ρ ) , Im λ < 0 , ρ ∈ R , (B.47) since it is sligh tly trickier. Here the transp osed op erators are defined as follows  Q t n − 1 ( λ ) φ  ( x n − 1 ) = Z R n − 1 d y n − 1 Q λ ( y n − 1 | x n − 1 ) φ ( y n − 1 ) , (B.48)  Λ t n ( − λ ) φ  ( x n − 1 ) = Z R n d y n Λ − λ ( y n | x n − 1 ) φ ( y n ) . (B.49) The k ernel of the ab o ve op erator pro duct equals Z R n − 1 d y n − 1 Z R n d z n Z R n d s n Q λ ( y n − 1 | x n − 1 ) Λ − λ ( z n | y n − 1 ) exp( − β e − z 1 ) × V ρ ( z n | s n ) Λ − ρ ( s n | t n − 1 ) . (B.50) Note that Λ − ρ ( s n | t n − 1 ), as a function of s n , b elongs to P n . So, again to prov e that the last in tegral is absolutely conv ergen t it is enough to show that the pro duct Q t n − 1 ( λ ) Λ t n ( − λ ) exp( − β e − x 1 ) V n ( ρ ) (B.51) is w ell defined on the space P n . F rom the explicit formulas ( B.7 ) we hav e Λ t n ( − λ ) exp( − β e − x 1 ) : φ ( x n ) 7→  Q n ( λ ) φ  (ln β , x n − 1 ) . (B.52) Besides, Q t n − 1 ( λ ) = I n − 1 Q n − 1 ( λ ) I n − 1 , I n − 1 : φ ( x 1 , . . . , x n − 1 ) 7→ φ ( − x n − 1 , . . . , − x 1 ) . (B.53) Therefore b y Prop osition B.1 and Lemma B.4 the pro duct ( B.51 ) is well defined on P n . 67 B.4 F rom equalit y of kernels to equality of op erators All lo cal relations b et ween GL and B C T o da op erators hold on the space of p olynomially b ounded con tinuous functions. In this section w e demonstrate that to establish the equality of op erators on this space it is sufficient to pro ve the equality of kernels. F or example, consider the commutativit y Q n ( λ ) Q n ( ρ ) = Q n ( ρ ) Q n ( λ ) , Im λ, Im ρ ∈ ( − g , 0) . (B.54) By Prop osition B.2 b oth sides of this relation are well defined on P n and, as w e argued in Section B.3 , the k ernels of b oth pro ducts are giv en by absolutely con vergen t in tegrals. In Section 2.4 using diagram tec hnique we prov e that these k ernels are equal to each other Z R n d y n Q λ ( x n | y n ) Q ρ ( y n | z n ) = Z R n d y n Q ρ ( x n | y n ) Q λ ( y n | z n ) . (B.55) Let us sho w that, as a consequence, the identit y ( B.54 ) holds on P n . Acting on φ ∈ P n from the left w e hav e  Q n ( λ ) Q n ( ρ ) φ  ( x n ) = Z R n d y n Z R n d z n Q λ ( x n | y n ) Q ρ ( y n | z n ) φ ( z n ) . (B.56) F rom definition   Q λ ( x n | y n )   =     Γ( g + Im λ ) Γ( g − i λ )     Q i Im λ ( x n | y n ) , (B.57) and clearly | φ | ∈ P n . Hence, by Prop osition B.2 the integral ( B.56 ) is absolutely con vergen t (in the initial order). T herefore, by F ubini–T onelli theorem w e can in terchange the order of in tegrals  Q n ( λ ) Q n ( ρ ) φ  ( x n ) = Z R n d z n  Z R n d y n Q λ ( x n | y n ) Q ρ ( y n | z n )  φ ( z n ) . (B.58) The same steps are applied to the right hand side of ( B.54 ). As a result, this iden tity follo ws from the equalit y of kernels ( B.55 ) in square brack ets. All other lo cal iden tities considered in the pap er hold on the space of p olynomially b ounded con tinuous functions by the same arguments. B.5 Bounds on B C T o da w av e function F or k n ∈ N n 0 denote standardly ∂ k n x n = ∂ k 1 x 1 · · · ∂ k n x n . (B.59) Also, for brevit y , we introduce C ( λ n ) = n Y j =1 1 | Γ( g − i λ j ) | . (B.60) In previous pap er [ BDK ] we prov ed the following b ound for B C T o da wa v e function. 68 Prop osition B.3. [ BDK , Pr op osition 1] L et k n ∈ N n 0 and x n , λ n ∈ R n . Then Ψ λ n ( x n ) is smo oth in x n and admits the b ound    ∂ k n x n Ψ λ n ( x n )    ≤ C ( λ n ) P ( | x 1 | , . . . , | x n | , | λ n | ) exp  k 1 (ln β − x 1 ) − β e − x 1  θ (ln β − x 1 ) + n − 1 X j =1  ( k j + k j +1 ) x j − x j +1 2 − e x j − x j +1 2  θ ( x j − x j +1 ) ! , (B.61) wher e P is p olynomial, whose c o efficients dep end on k n , β , g . In p articular, for k n = (0 , . . . , 0) it do esn ’t dep end on λ n   Ψ λ n ( x n )   ≤ C ( λ n ) P ( | x 1 | , . . . , | x n | ) × exp − β e − x 1 θ (ln β − x 1 ) − n − 1 X j =1 e x j − x j +1 2 θ ( x j − x j +1 ) ! . (B.62) In Section 2.8 to pro ve the equiv alence of Gauss–Giv ental and Mellin–Barnes represen tations for B C T o da wa v e function we use the inv ersion formula for GL T o da wa v e function. According to W allac h [ W1 ] (see also [ W2 , Section 7]), it holds for the functions from Whittaker Schwartz sp ac e T , which consists of ψ ( x n ) ∈ C ∞ ( R n ) suc h that sup x n ∈ R n exp n − 1 X j =1 m j x j − x j +1 2 ! (1 + | x n | ) d |D ψ ( x n ) | < ∞ , (B.63) for all m j , d ∈ N 0 and constan t co efficien t differential op erators D . F rom the ab o ve prop osition w e hav e the follo wing statement. Corollary B.2. L et λ n ∈ R n and ε > 0 . Then e − ε x n − β e − x 1 Ψ λ n ( x n ) ∈ T . (B.64) Pr o of. F or brevity , denote function in question ψ ( x n ) = e − ε x n − β e − x 1 Ψ λ n ( x n ) . (B.65) First, by Prop osition B.3 it is smo oth. Second, let us calculate its deriv ativ es. F or any k n ∈ N n 0 w e hav e ∂ k n x n ψ ( x n ) = X i c i ( ε ) e ℓ ( i ) (ln β − x 1 ) − ε x n − β e − x 1 ∂ s ( i ) n x n Ψ λ n ( x n ) , (B.66) with some co efficien ts c i and in tegers ℓ ( i ) ∈ N 0 , s ( i ) n ∈ N n 0 , such that ℓ ( i ) ≤ k 1 , s ( i ) j ≤ k j . Since ℓ ( i ) ≥ 0, e ℓ ( i ) (ln β − x 1 ) − β e − x 1 ≤ e [ ℓ ( i ) (ln β − x 1 ) − β e − x 1 ] θ (ln β − x 1 ) . (B.67) 69 Hence, using Prop osition B.3 and the fact that ℓ ( i ) ≤ k 1 , s ( i ) j ≤ k j w e derive estimate   ∂ k n x n ψ ( x n )   ≤ P ( | x 1 | , . . . , | x n | ) exp − ε x n + 2  k 1 (ln β − x 1 ) − β e − x 1  θ (ln β − x 1 ) + n − 1 X j =1  ( k j + k j +1 ) x j − x j +1 2 − e x j − x j +1 2  θ ( x j − x j +1 ) ! , (B.68) whic h is sufficient to prov e ( B.63 ). Notice that function (1 + | x n | ) d from ( B.63 ) can b e absorb ed in to p olynomial P (1 + | x n | ) d P ( | x 1 | , . . . , | x n | ) ≤ (1 + | x n | 2 ) d P ( | x 1 | , . . . , | x n | ) , (B.69) while in the exp onen t from ( B.63 ) w e can add step functions exp n − 1 X j =1 m j x j − x j +1 2 ! ≤ exp n − 1 X j =1 m j x j − x j +1 2 θ ( x j − x j +1 ) ! , (B.70) so that it has the same form, as exp onen ts from the third line ( B.68 ). So, these tw o factors do esn’t essentially change estimate from the righ t hand side ( B.68 ). T o see that this estimate is b ounded for all x n ∈ R n , w e pass to the v ariables y 1 = ln β − x 1 , y 2 = x 1 − x 2 2 , . . . y n = x n − 1 − x n 2 . (B.71) Expanding p olynomial P in monomials | y 1 | d 1 · · · | y n | d n w e can rewrite the estimate ( B.68 ) as the sum of factorised functions | y 1 | d 1 e nε y 1 +2[ k 1 y 1 − e y 1 ] θ ( y 1 ) n Y j =2 | y j | d j e 2( n +1 − j ) εy j +[( k j + k j +1 ) y j − e y j ] θ ( y j ) , (B.72) Since ε > 0, these functions are b ounded for all y n ∈ R n . C Mellin–Barnes b ounds and analyticit y In this section we prov e that Mellin–Barnes represen tations for GL and B C wa v e functions are absolutely conv ergen t and entire in sp ectral parameters. F or this w e use b ounds on gamma functions | Γ( a + i b ) | ≤ √ 2 π e 1 6 a | a + i b | a − 1 2 e − π 2 | b | , (C.1) 1 | Γ( a + i b ) | ≤ e a + 1 6 a √ 2 π | a + i b | − a + 1 2 e π 2 | b | , (C.2) where a > 0, b ∈ R . These b ounds follow from [ PK , p. 34]. 70 C.1 GL system The Mellin–Barnes represen tation for GL T o da wa ve function is defined by recursiv e formula Φ λ n ( x n ) = Z ( R +i r ) n − 1 d γ n − 1 ˆ µ ( γ n − 1 ) e i( λ n − γ n − 1 ) x n n Y j =1 n − 1 Y k =1 Γ(i λ j − i γ k ) Φ γ n − 1 ( x n − 1 ) (C.3) with one particle w av e function Φ λ 1 ( x 1 ) = e i λ 1 x 1 and measure ˆ µ ( λ n ) = 1 n ! (2 π ) n n Y j,k =1 j  = k 1 Γ(i λ j − i λ k ) = 1 n ! (2 π ) n Y 1 ≤ j Im λ j for all j = 1 , . . . , n . The analytic contin uation of wa ve function in sp ectral v ariables is achiev ed b y shifting in- tegration contours. T o prov e that the ab o ve in tegral is absolutely con vergen t and, as a conse- quence, justify suc h shifts, we need the following inequality . Lemma C.1. [ BDKK , L emma 2] L et ( y m 1 , . . . , y mm ) ∈ R m for m = 1 , . . . n . Then for any δ > 0 ther e exists ε > 0 such that 2 n − 1 X m =2 X 1 ≤ j 0 we have   Φ λ n ( x n )   ≤ P δ ( λ n ) exp  − π 2 X 1 ≤ j r 2 > . . . > r n − 1 ≡ r > Im λ j . F rom definition ( C.4 ) we hav e b ound on the measure   ˆ µ ( γ m )   ≤ P ( γ m ) exp  π X 1 ≤ j 0 this function is b ounded by P (Re γ 1 , . . . , Re γ n − 1 , Re λ n ) exp  n − 1 X m =1 ( x m +1 − x m ) mr m − x n n X j =1 Im λ j − π 2 X 1 ≤ j 0. This gives con v ergence uniform in λ j from compact subsets. Besides, this estimate ensures that we can shift integration contours. Hence, for λ n ∈ R n w e can send r m → 0 (one by one), whic h gives the claimed b ound ( C.7 ). C.2 B C system The first Mellin–Barnes represen tation for B C wa ve function is given b y Ψ λ n ( x n ) = e β e − x 1 Z ( R − i r ) n d γ n ˆ µ ( γ n ) (2 β ) − i γ n n Q j,k =1 Γ(i γ j ± i λ k ) Q 1 ≤ j 0 is chosen so that p oles of the in tegrand γ k = ± λ k + i m, m ∈ N 0 (C.13) are ab ov e the integration contours, that is r > | Im λ j | for j = 1 , . . . n . As b efore, to analytically con tinue this function in sp ectral parameters w e shift con tours, which is justified by the follo wing prop osition. Prop osition C.2. L et | Im λ j | < r for j = 1 , . . . , n . Then the inte gr al ( C.12 ) is absolutely c onver gent uniformly in λ j fr om c omp act subsets. Corollary C.2. The function Ψ λ n ( x n ) c an b e analytic al ly c ontinue d to λ n ∈ C n . 72 Pr o of of Pr op osition C.2 . First, change in tegration v ariables γ j = ρ j − i r , so that ρ j ∈ R , and use the relation Φ γ n ( x n ) = e r x n Φ ρ n ( x n ) . (C.14) Then with the help of inequalities ( C.1 ), ( C.2 ), ( C.7 ) and ( C.9 ) we estimate the absolute v alue of the in tegrand in ( C.12 ) by the function f ( λ n , ρ n ) exp  π 2 X 1 ≤ j | Im λ j | and r < g . Its conv ergence can b e prov en in analogous wa y , but this time we ha ve analytic contin uation only to the strip | Im λ j | < g . Prop osition C.3. L et | Im λ j | < r < g for j = 1 , . . . , n . Then the inte gr al ( C.19 ) is absolutely c onver gent uniformly in λ j fr om c omp act subsets. 73 D Gustafson in tegral reduction The follo wing identit y is pro ved in [ G , Theorem 9.3] 1 (4 π ) n n ! Z R n d λ n 2 n +2 Q j =1 n Q k =1 Γ( z j ± i λ k ) Q 1 ≤ j 0 for j = 1 , . . . , 2 n + 2. Let us pro ve that in the limit z 2 n +2 → ∞ it reduces to the form ula 1 (4 π ) n n ! Z R n d λ n 2 n +1 Q j =1 n Q k =1 Γ( z j ± i λ k ) Q 1 ≤ j

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Leave a Comment