Singular Soliton Operators and Indefinite Metrics

P .Grinevic h, S.No vik o v 1 , 2 Singular Soliton Op erators and Indefinite Metr i cs Abstract . We c onsider singular r e al se c ond or der 1D Sc hr¨ odinger op er ators such that al l lo c al solutions to the eigenvalue pr oblem s ar e x -mer omorphic for al l λ . A l l alg e b r o-ge om etric al p otentials (i.e. “sin g ular finite-gap” and “singular solitons”) satisfy to this c ondition. A Sp e ctr al The ory is c onstructe d for the p erio dic and r apid ly de c r e asing p otentials in the class e s of functions with singularities: The c o rr esp ondin g op er ators ar e s ymmetric with r esp e ct to some natur al indefinite i n ner pr o duct as it was disc over e d by the pr esent authors. It has a finite n umb er of ne gative squar e s in the b oth (p erio dic and r apid ly de cr e asing) c ases. The time dynamics pr ovide d by the KdV hier ar chy pr eserves this numb er. The rig ht analo g of F ourier T r ansform on Riemann Surfac es with go o d multiplic ative pr op erties (the R-F o urier T r ansform) is a p artial c ase of this the ory. The p otential has a p ole in this c ase at x = 0 with asymptotics u ∼ g ( g + 1) /x 2 . Her e g is the gen us of sp e ctr al curve. 1 P .G.Gr inevich, Landau Institute fo r Theo retical Physics, Moscow, Russia e-mail pg g@landau.a c.ru, S.P .Noviko v, Universit y of Ma ryland at Co llege Park, USA, a nd Landau Institute for Theoretical P hysics, Moscow, Russia e-mail noviko v@ipst.umd.edu 2 This work was partia lly supp orted by the Russian F ounda tio n for Basic Research, grant 11-01- 0 0197 -a. The first author was a lso partially suppo r ted by the Russian F e der - ation Gov ernment gr ant No 2 010-2 20-01 -077, b y the pr ogra m “Lea ding scient ific schools” (grant NSh-4995.2 012.1) and by the program “ F undamental pr oblems of nonlinear dynam- ics” 1 The Main Con structions and Results The F ourier T ransform and Riemann Surfaces Consider real ∞ -smo ot h p oten tials u ( x ) meromorphic in some small com- plex area near the p o int x j ∈ R as in [1, 2]. The following Statemen t 1 can b e easily prov ed. 3 Statemen t 1. Al l solutions to the Sturm-Liouvil le e quations (for al l λ ) L Ψ = − Ψ ′′ + u ( x )Ψ = λ Ψ ar e mer omorphic in some smal l nei g h b ourho o ds of the p oints x j if and only if u ( x ) = n j ( n j + 1) / ( x − x j ) 2 + n j − 1 X k =0 u j k ( x − x j ) 2 k + O  ( x − x j ) 2 n j  for s o me inte ger n j ∈ Z . Ther e exists a b asis of solutions ne ar x j such that for y = x − x j and al l λ ψ 1 j = 1 /y n j + a 1 ( λ ) /y n j − 2 + a 2 ( λ ) /y n j − 4 + . . . + a n j ( λ ) /y − n j + O ( y n j +1 ) (1) ψ 2 j = y n j +1 + . . . Statemen t 2. A l l algebr o ge ometric (A G) p otentials satisfy to the c onditions of the Statement 1. By definition for ev ery A G op erat o r L there exists a linear differential op erator A of an o dd or der suc h that [ L, A ] = 0. It is w ell-kno wn that all eigenfunctions are x - meromor phic (see [3]). W e call suc h p oten tials u ( x ) “singular finite-g ap” if they are p erio dic in x : u ( x + T ) = u ( x ). W e call them “singular solitons” if u ( x ) → 0, | x | → ∞ . The simples t examples kno wn in the classical literature are “the singular solitons”: u ( x ) = n ( n + 1) x 2 , u ( x ) = n ( n + 1) k 2 sinh 2 ( k x ) 3 Essentially , this Statement can b e found in the pa p e r [1 5]: page 16 9, Pro po sition 3.3. The corres p o nding Corolla ry for KdV equation w as prov ed in this pap er for rational po tent ials only . It is also true for a ll finite-ga p p otentials. 2 and “the Lam´ e p otentials” (degenerate and non- degenerate) u ( x ) = n ( n + 1) k 2 sin 2 ( k x ) , u ( x ) = n ( n + 1) ℘ ( x ) The “Diric hlet Problem” for the real Lam´ e p oten tials at the in terv al [0 , T ] with real perio d T = 2 ω and imaginary p erio d T ′ = 2 ω ′ w as studied b y Hermit. No sp ectral theory for the L a m ´ e op erators on the whole line R has b een studied in the classical literature. W e are going t o construct a sp ectral t heory for the operator L in some space of functions on the real line w ith Indefinite Inner Pr o duct. Let us describ e this space. Fix a set X of p oin ts x j ∈ R , j = 1 , . . . , N and n um b ers n j ∈ Z + . This set should b e finite for the class of rapidly decreasing p oten tials u ( x ) = O  1 /x 2  , | x | → ∞ . In the perio dic case its in tersection with any p erio d ( x, x + T ) should b e finite 4 . F or the p erio dic case w e fix also some unitary Blo c h multiplier κ , | κ | = 1, where Ψ( x + T ) = κ Ψ( x ) . W e c ho ose class of functions F 0 X , ∞ -smo o t h outside of the p oints x j (and their p erio dic shifts), suc h t ha t near ( x j ) w e hav e Ψ( y ) + ( − 1) n j +1 Ψ( − y ) = O ( y n j +1 ) , (2) y = x − x j , j = 1 , . . . , N . The whole sp ace of functions F X ∋ F 0 X consists of functions Ψ with ”principal parts” Φ j at the p oints x j ∈ X Φ j ( y ) = n j X k =0 a j k /y n j − 2 k , y = x − x j . (3) Ψ = Φ j + O  y n j +1  , so the difference Ψ − Φ j satisfies the defining conditions F 0 X lo cally at x j for all j = 1 , . . . , N . Ev en more, this difference has the order O ( y n j +1 ) at x j . 4 There is a very sp ecial interesting case where all x j = j T , j ∈ Z . In par ticular, the genus of sp ectra l curve is exactly equal to n j = g for the famous Lam´ e p otentials. 3 The standard inner pro duct < Ψ 1 , Ψ 2 > = T Z 0 Ψ 1 ( x ) Ψ 2 ( x ) dx, u ( x + T ) = u ( x ) , or < Ψ 1 , Ψ 2 > = ∞ Z −∞ Ψ 1 ( x ) Ψ 2 ( x ) dx, u ( x ) → 0 , | x | → ∞ can b e extended to the class F X b y means of the fo rm ula < Ψ 1 , Ψ 2 > = T Z 0 Ψ 1 ( x ) Ψ 2 ( ¯ x ) dx, u ( x + T ) = u ( x ) , or (4) < Ψ 1 , Ψ 2 > = ∞ Z −∞ Ψ 1 ( x ) Ψ 2 ( ¯ x ) dx, u ( x ) → 0 , | x | → ∞ (there appears the ¯ x instead of x in the se cond factor of the inte grand compared with the previous t w o expressions) a nd av oiding singular p oin ts through the complex domain. Our requiremen ts imply the fo llo wing: The pro duct Ψ 1 ( x ) Ψ 2 ( ¯ x ) is x -meromorphic. Its residues near the singularities are equal to zero. So our inner pro duct is we ll-defined (but indefinite) 5 . Statemen t 3. The in ner pr o duct (4) is wel l- d efine d after this r e gularization. It is indefinite with exactly l X = N P j =1 l n j ne gative squar es for e ach κ ∈ S 1 wher e l n j =  n j + 1 2  The pro duct is p ositiv e in the subspace F 0 X ∈ F X b y definition. Ev ery co efficien t a j k , k = 0 , . . . , l n j − 1 (i.e. corresp o nding to the negativ e p ow ers of 5 In our r ecent pap er s [16], [17] w e hav e shown that this scalar pro duct is w ell-defined on the eigenfunctions of forma lly symmetric real finite-gap op era tors of arbitrar y order and on the eig enfunctions of the non- stationary Schr¨ odinger o per ators with one spa tial v aria ble. 4 y ) giv es exactly one negativ e square. The p ositiv e p ow ers of y do not destroy p ositivit y of inner pro duct. Detailed pro of of the Statemen t 3 is presen ted in the App endix 2 . Let Γ b e a real h yp erelliptic Riemann Surface w 2 = R 2 g +1 ( z ) of the Blo ch- Flo quet function Ψ ± ( x, z ) in the “finite-gap” p erio dic case. It has exactly t w o an tiholomorphic in v olutions τ ± where z → ¯ z . W e ch o ose τ = τ + suc h that the “infinite cycle” ( i.e. the sp ectral zone where z ∈ R and z → + ∞ ) b elongs to the fix p oint set. The z - p oles of Ψ do not dep end on x . They fo r m a divisor D consisting of g p oin ts. Here g is the gen us of Γ. By definition t he canonical con tour p 0 ∈ Γ consist of all p oints with unimo dular multipliers | κ | = 1 (see[1]) . In the decomp osition theorem b elo w we assume that it is nonsingular. The p 0 is in v ariant under the a ction of antiin v olution τ . The infinite comp onen t of the canonical con tour contains an infinite p oint ∞ ∈ Γ. The antiin v olution τ a cts trivially on the real part of that comp onent. By fixing κ w e get a coun table set of p oints z q = ( λ q ( κ ) , ± ) in the canonical con tour. Let us consider t he corresp onding set of functions Ψ q = Ψ( x, z q ). Except o f finite num b er, all these p oin ts b elong to the infinite comp onen t . Our Sp ectral T ransform maps the space of C ∞ -functions on the canonical con tour ( pro p erly decreasing at infinit y) in to t he space of functions F X on the real line R . It preserv es a n indefinite metric as it was pro v ed in [1]. In the presen t w ork w e describe the image of this T ransform. It is the whole space F X . Let us describ e first the case of smo oth real p erio dic op erators: The set X is empty . All branc hing p oin ts o f the Riemann surface a re real. The divisor D con tains exactly one simple p ole in eac h finite gap cycle (see [3]. The union of all gaps is exactly equal to the fixpoint set of the second antiin v olution τ − . The Riemann analog of the F ourier T ransform (the R-F ourier T ransform) corresp onds to the case of real Riemann surfaces but some branch ing p oints ma y b e complex. The divisor should b e c on- cen trated at the infinite p oint D = g × ∞ . The Bak er–Akhiezer family of functions ψ x ( γ ) = Ψ( x, γ ) has the b est p ossible m ultiplicativ e prop erties in this case: they are similar to the pro p erties o f the standard exp o nen t ial basis of the ordina r y F ourier T ransform where the genus zero surface is w 2 = z , and the canonical con tour is an infinite cycle o v er the real p ositiv e half-line. F or the R-F ourier T r a nsform case there exists a singular p oint x j = 0 in the rapidly decre asing case, and a singular p oint for ev ery p erio d in the p erio dic 5 case. It is such that n j is equ al to the genus 6 . T here exists an op erato r R = ∂ g x + a 1 ∂ g − 1 x + ... with co efficien ts dep ending on x, y only , such that Ψ( x, P )Ψ( y , P ) = R Ψ( x + y , P ) . W e hav e a 1 = − ( ζ ( x + ζ ( y ) − ζ ( x + y ))) for g = 1 and Lam ´ e p oten tial (the Hermit case). It is easy to describe the co efficien ts fo r all R iemann surfaces. The Riemann analo g of F ourier Series with go o d m ultiplicativ e prop erties w as dev elop ed b y Kriche v er and No vik o v in the series of w orks made in the late 1980 s. They dev elop ed the op erator construction of the bo sonic (P oly ak o v type) string theory for all diagrams whic h are the Riemann surfaces of all genera (see in the b o ok [6]). No analog of indefinite inner pro duct has b een discussed. Theorem 1. Every function f ∈ F X such that f ( x + T ) = κ f ( x ) c an b e uniquely pr esente d in the form f = X q c q Ψ q , λ q = λ q ( κ ) , c q = < f , Ψ q > / < Ψ q , Ψ q > . wher e L Ψ q = λ q Ψ q , Ψ q = Ψ( x, z q ) , z q = ( λ q ( κ ) , +) or z q = ( λ q ( κ ) , − ) , and u ( x ) is a r e al p erio dic s i n gular finite-gap p otential. This series c onver ges in the sense that the c o efficients of the sing ular p arts do c on ver ge (mor e r apid ly than any p ower), and in so me neighb orho o d of the p oints x j the series P q (Ψ q − Φ q j ) c q c onver g es to the c o rr esp ond i n g differ enc es (Ψ − Φ j ) with al l derivatives, ne ar every p oint x j . Her e Φ q j = n j − 1 P q =0 a ( q ) j k /y n j − 2 k , y = x − x j , which a r e the singular p arts o f the eigenfunctions Ψ q , at the p oints x j ∈ X , and Φ j is the singular p a rt of f ∈ F X as it was define d ab ov e in the F o rm ula 3. Theorem 1’. Cons i d er a r apid ly de cr e a s i n g p o tential u ( x ) . F or every func- tion f ∈ F X de cr e asing r api d ly enough at | x | → ∞ , we have the fol lowing r epr esen tation f = Z k ∈ R c k Ψ k ( x ) dk + X m d m Ψ m , L Ψ k = k 2 Ψ k , L Ψ m = λ m Ψ m , 6 In the famous cases o f the Lam´ e p otentials there exists only one singula r po in t at the per io d. W e in vestigate in App endix 3 how ma ny additional singula r ities of the s maller t yp es might app ear in the R-F ourier T ra nsform case. 6 Her e u ( x ) = O  1 /x 2  at | x | → ∞ , and w e as s ume that u ( x ) is a si n gular multisoliton p otential. Remark. Recen tly the autho rs ha v e pro v ed, that decomp osition for mu- las from Theorem 1 and Theorem 1’ are v alid for all p erio dic finite-gap real and complex p oten tials (they may b e regular or singular). The pro of is based o n the reduction to the regular complex case a nd follo ws the same sc heme as used in Theorems 1 and 1’. Suc h results for the standard po sitive Hilb ert spaces and regular self- adjoin t 1D stationary Sc hr¨ odinger operato r s where kno wn ma ny y ears at the folklore lev el (see the form ulas and quotations in the a rticle [3]). F or more complicated situation of non-stationary 1D Sc hr¨ odinger op erato rs and sta- tionary 2D Sc hr¨ odinger op erato r s the sp ecific finite-gap for m ulas and decom- p osition theorems on R iemann surfaces w ere obtained in the or ig inal w orks [4], [5]. In our indefinite case, w e use essen tially the same tec hnique. Our program is to extend t hese r esults to the whole c lass of p erio dic and rapidly decreasing infinite-gap real p erio dic p oten tials with singularities of the type describ ed ab ov e. By the w a y , in the work [7] the “scattering dat a” we re constructed for the case u ( x ) = O  1 /x 2  at | x | → ∞ , all n j = 1. Indefinite metric, sp ectral theory and decomp osition of f unctions w ere not discussed in t his w ork. The theory of our functional spaces is based on the solution of the fol- lo wing problem: consider the KdV solutions u t = 6 uu x − u xxx suc h that u ( x, 0) = n ( n + 1) /x 2 . It is well-kno wn, that w e can write these solutions in the form u ( x, t ) = 2 n ( n +1) / 2 X q =1 1 ( x − x q ( t )) 2 , It is easy to see, that x q = a q t 1 / 3 . How many of x j -s are real? Though there is a huge a moun t of literature dedicated to the ra tional, tr ig onometric and elliptic solutions t o t he K dV hierarch y , w e could not find the lemma b elo w an ywhere. So w e prov ed it ourselv es. Statemen t 4. Exactly l n =  n +1 2  p oles r em ain r e al. This numb er is exactly e qual to the numb er of c o efficients a k j at every sing ular p oint x j with n j = n . Remark The following transfor ma t io ns preserv e the set { a q } : a q → ¯ a q , a q → ξ a q , ξ 3 = 1. Pro of of Statemen t 4. It is clear, that this problem is equiv alen t t o the following one: consider the K dV solutions u t = 6 uu x − u xxx suc h, that 7 u ( x, 0) = n ( n + 1) ℘ ( x ), where ℘ is the W eierstrass function, asso ciat ed with a real rectangular lattice. Ho w man y p oles remain at the real p erio d for t > 0, t ≪ 1? Let us assume that some generic unitary Bloch multiplier κ 0 is fixed. It follows from the App endix 2 that the space F X has exactly l n =  n +1 2  negativ e squares for t = 0 . W e use no w the Theorem 1 prov ed in the Ap- p endix 1: An y collection of singularities can b e a pproximated by t he image of the F ourier ma p. Therefore the n um b er of negative squares is equal to the num b er of p oin ts γ at the canonical con tour suc h that exp( ip ( γ ) T ) = κ 0 and dµ ( γ ) /d p ( γ ) < 0. Here dµ is the sp ectral measure in the decomp osition form ula dµ = ( λ ( γ ) − λ 1 ) . . . ( λ ( γ ) − λ g ) 2 p ( λ ( γ ) − E 0 ) . . . ( λ ( γ ) − E 2 g )) dλ ( γ ) . This n umber do es not dep end on t . Therefore the num b er of negative squares in the metrics in F X also do es no t dep end on t . F or small t > 0 all singu- larities ar e simple. Therefore the n um b er of nega t ive squares coincides with the n um b er of real singular p o in ts on the p erio d. This completes the pro of. Let us p oin t out, that we already prov ed in the w ork [1] that l ′ n ≥ l n . Here l ′ n is the num b er of real a q . The l n is equal to the num b er of negat ive squares in the inner pro duct ab ov e fo r this sp ecific case. This quantit y is time-in v ariant. A naive understanding of the oppo site inequalit y l ′ n ≤ l n is the following: as numerical calculations sho w, the p oin ts x q ( t ) for small t > 0, t ∈ R , are lo calized appro ximately in t he p oints of equilateral triang le (see Fig 1). 8 n=4 l=2 n=5 l=3 n=1 l=1 n=2 l=1 n=3 l=2 Fig 1. The p oles a q for differen t v alues of n . F or the case of the ideal equilateral tr ia ngle w e ob viously ha v e l ′ n = l n =  n +1 2  . Ho w ev er, in fa ct, it is sligh tly perturb ed. So w e should hav e l ′ n ≤ l n if p erturbation is really small. But the symmetry a q → ¯ a q k eeps all real p oin ts on the real line. So w e a re done with the really small p erturbations of the equilateral triangle. But our p erturbation is only numeric ally small, not theoretically . So this a rgumen t is non- rigorous. Remark. The p ositions of these zero es w ere studied n umerically and analytically in [8]. Ho w ev er, the problem of calculation of the n um b er of real zero es w as not discussed in [8]. It is not clear whether it is p o ssible to obtain rigorous pro of of our result based on the estimates fro m t his pa p er. The first rigorous pro of of the inequalit y l ′ n ≤ l n w as completed with the help of our studen t A.F etiso v. It is different from the pro of presen ted ab ov e. A non-standard example w e obtain for the case of elliptic f unction u ( x ) = 2 ℘ ( x ) corresponding to the rhom bic lattice ( see Fig. 2 a. Fig. 2b). The canonical contour is connected in this case. It has tw o singular p oin ts. The an tiin v olution is not equal to iden tit y at the contour in t his case, so there are no self-adjoin t real problems on t he real line for suc h R iemann surface. The inner pro duct is alw ay s indefinite. The pro jection of the con tour t o the 9 plane of the spectral parameter con tains a complex part, so the sp ectrum of the op erato r is complex for such real singular finite-g ap p oten tial. E 1 E 2 E 3 Fig 2a The rhombic lattice Fig 2b The contour | κ | = 1 is singular The classical Lam´ e problems do not lead to this case, so it nev er was considered. App endix 1. Pro of of Theore m 1 W e prov e Theorem 1 here and construct an analog of the con tin uous F ourier decomp osition b y the eigenfunctions of singular finite-g ap op erator with a p erio dic p oten tial. The pro of consists of tw o steps. 1. W e reduce decomp osition problem for the real singular p otential to the decomp osition problem for regular complex p oten tials. 2. W e construct eigenfunction expansion for r egular complex p oten tials. 10 1.1 Notations Let us recall some basic definitions. The sp ectral curv e Γ is defined by : µ 2 = ( λ − E 0 ) . . . ( λ − E 2 g ) = R ( λ ) . Our divisor is D = γ 1 + . . . + γ g . The follo wing no tations will b e us ed in Appendix: λ ( γ ) denotes the pro jection o f the p o int γ ∈ Γ t o the λ -plane. So either γ = ( λ ( γ ) , +) or γ = ( λ ( γ ) , − ). Let λ 1 = λ ( γ 1 ),. . . , λ g = λ ( γ g ). The quasimomen tum differen tia l d p is uniquely defined by the following prop erties: 1. dp is holo mo r phic in Γ outside the p oint λ = ∞ . 2. dp = dk  1 + O  1 k 2  , k 2 = λ near the p oint λ = ∞ . 3. In tegrals ov er all basic cycles are purely real Im I c dp = 0 (5) for an y closed con tour c ⊂ Γ The quasimomen tum function p ( γ ) is the primitiv e of dp , and it is alw a ys m ultiv alued. W e assume, tha t p ( γ ) = k + O  1 k  . F rom (5) it follo ws, that the ima g inary part of the quasimomen tum func- tion Im p ( γ ) is w ell-defined. W e assume, t ha t our potential u ( x ) is p erio dic with the p erio d T . It implies, that exp( ip ( γ ) T ) is a single-v alued function in Γ. 11 As ab ov e, w e denote the Blo c h function b y Ψ( γ , x ), and σ denotes the holomorphic inv olution, in terc hanging the sheets o f the surface Γ: σ : ( λ, +) → ( λ, − ) , Ψ ∗ ( γ , x ) = Ψ( σ γ , x ) Assume, that function f ( x ) has finite supp ort and f ∈ F X . Let us define the con tinu ous F ourier transform for f ( x ) by : ˆ f ( γ ) = 1 2 π ∞ Z −∞ Ψ ∗ ( y , γ ) f ( y ) dy (6) where w e use t he r ule o f g o ing a round the singularities within the complex domain. Theorem 1”. Assume, that: 1. The sp e ctr al curve Γ is r e gular (has no multiple p oints), 2. The c ontour Im p ( γ ) = 0 is r e gular, i.e. dp ( γ ) 6 = 0 eve rywher e at this c ontour. Then we hav e the fol lowin g r e c onstruction form ula: f ( x ) = I Im p ( γ )=0 ˆ f ( γ )Ψ( x, γ ) ( λ ( γ ) − λ 1 ) . . . ( λ ( γ ) − λ g ) 2 p ( λ ( γ ) − E 0 ) . . . ( λ ( γ ) − E 2 g )) dλ ( γ ) . (7) F o r any r e gular x the inte gr and in (7) de c ays for γ → ∞ faster, than any de gr e e of λ . 1.2 Reduction to smo oth p oten tial Apply now a series of Crum transformat ions. W e in tend to reduce the de- comp osition with resp ect to singular real p oten tial to the decomp osition with resp ect to complex nonsingular p oten tial. Lemma 1. L et n = n max denotes the m a ximal or der o f singularity n max = max j n j . Th e n ap p lying a series of n pr o p erly c hosen Darb oux–Crum tr anf o r- mation one c an obtain a r e gular complex p o tential. 12 Pro of. Consider t he image of all divisor tra jectories in Γ. The p otential u ( x ) is p erio dic. Therefore they form a compact set. Let q 1 ∈ Γ b e a p oin t outside of this set, suc h that Im p ( q 1 ) 6 = 0, l 1 = λ ( q 1 ). Let ψ 1 ( x ) = Ψ( q 1 , x ) b e the corresp onding Bak er–Akhiezer function. Lψ 1 ( x ) = l 1 ψ 1 ( x ) . Then 1. ψ 1 ( x ) = 1 ( x − x j ) n j ( a ( j ) 0 + o (1)) , a ( j ) 0 6 = 0 at all singular p oints. 2. ψ 1 ( x ) 6 = 0 for all x ∈ R , x 6 = x j . Let Q 1 =  ∂ x − ψ x ψ  , Q ∗ 1 =  − ∂ x − ψ x ψ  . W e hav e L − l 1 = Q ∗ 1 Q 1 , L 1 − l 1 = Q 1 Q ∗ 1 , where L 1 denotes the (D arb oux-Crum)-tra nsfor med op erator with p otential u (1) ( x ) = u ( x ) − 2 ∂ 2 x log( ψ 1 ) and Blo c h function Ψ (1) ( x, γ ) = 1 λ − l 1 Q 1 Ψ( x, γ ) . W e see, that this t ransformation reduces the orders o f all singularities by 1 and generates no new singular p oints . By rep eating t his pro cedure n times w e come to the smo oth p oten tial u ( n ) ( x ). Let us define the o p erators L = L 0 , L 1 , . . . , L n b y the follo wing form ulas L n = − ∂ 2 x + u ( n ) ( x ) , L k − l k = Q k Q ∗ k , L k − l k +1 = Q ∗ k +1 Q k +1 , Q k Q ∗ k = Q ∗ k +1 Q k +1 + l k +1 − l k This pro cedure generates Blo ch functions with slightly non-standard nor- malization. T o obtain the standard Bak er–Akhiezer f unction, it is necessary to c hange normalizatio n of Ψ( x, k ). Let us denote by γ 1 ( x ), . . . , γ g ( x ) the divisor of zero es of Ψ( x, γ ). 13 Let x 0 b e one of the singular p oin ts with the highest order singularity . It means that for x = x 0 exactly n p oints of the divisor γ 1 ( x 0 ), . . . , γ g ( x 0 ) are lo cated at the p oint λ = ∞ . Denote the remaining p oin ts b y γ 1 ( x 0 ), . . . , γ g − n ( x 0 ). Let ˜ Ψ( x, γ ) b e the Bak er–Akhiezer function with g − n simple p oles γ 1 ( x 0 ), . . . , γ g − n ( x 0 ) at the finite part of Γ. It has the asymptotics ˜ Ψ( x, γ ) = e ik ( x − x 0 ) (( − ik ) n + O ( k n − 1 )) , k 2 = λ, λ → ∞ . Then ˜ Ψ ( n ) ( x, γ ) = 1 ( λ − l 1 ) · · · ( λ − l n ) · Q n Q n − 1 . . . Q 1 ˜ Ψ( x, γ ) (8) is the Bak er–Akhiezer function for the smo oth op erator L n with the divisor of p oles γ 1 ( x 0 ), . . . , γ g − n ( x 0 ), σ q 1 , . . . , σ q n and essen tial singularit y ˜ Ψ ( n ) ( x, γ ) = e ik ( x − x 0 ) (1 + o (1)) , λ → ∞ . (9) Lemma 2. The op er ators Q j , Q ∗ J map Blo ch functions to the Blo ch f unction s with the same multiplier. Consider the fo llowing op erator: M = Q n · . . . · Q 1 · Q ∗ 1 · . . . · Q ∗ n Lemma 3. We h ave the fol low i n g formula M = ( L n − l 1 )( L n − l 2 ) . . . ( L n − l n ) The pro of is straigh tforw ard: Q n . . . Q 4 Q 3 Q 2 Q 1 Q ∗ 1 Q ∗ 2 Q ∗ 3 Q ∗ 4 . . . Q ∗ n = Q n . . . Q 4 Q 3 Q 2 ( Q ∗ 2 Q 2 + l 2 − l 1 ) Q ∗ 2 Q ∗ 3 Q ∗ 4 . . . Q ∗ n = = Q n . . . Q 4 Q 3 ( Q 2 Q ∗ 2 + l 2 − l 1 ) Q 2 Q ∗ 2 Q ∗ 3 Q ∗ 4 . . . Q ∗ n = = Q n . . . Q 4 Q 3 ( Q ∗ 3 Q 3 + l 3 − l 1 )( Q ∗ 3 Q 3 + l 3 − l 2 ) Q ∗ 3 Q ∗ 4 . . . Q ∗ n = = Q n . . . Q 4 ( Q 3 Q ∗ 3 + l 3 − l 1 )( Q 3 Q ∗ 3 + l 3 − l 2 ) Q 3 Q ∗ 3 Q ∗ 4 . . . Q ∗ n = = Q n . . . Q 4 ( Q ∗ 4 Q 4 + l 4 − l 1 )( Q ∗ 4 Q 4 + l 4 − l 2 )( Q ∗ 4 Q 4 + l 4 − l 3 ) Q ∗ 4 . . . Q ∗ n = = Q n . . . ( Q 4 Q ∗ 4 + l 4 − l 1 )( Q 4 Q ∗ 4 + l 4 − l 2 )( Q 4 Q ∗ 4 + l 4 − l 3 ) Q 4 Q ∗ 4 . . . Q ∗ n = . . . = ( Q n Q ∗ n + l n − l 1 )( Q n Q ∗ n + l n − l 2 ) . . . ( Q n Q ∗ n + l n − l n − 1 ) Q n Q ∗ n = = ( L n − l 1 )( L n − l 2 ) . . . ( L n − l n − 1 )( L n − l n ) 14 Corollary 1. The op er ator M is a diff e r ential op e r ator with smo oth c o effi- cients. Remark 1. It fol lows fr om the de fi nition of Q k that Q ∗ 1 · . . . · Q ∗ n · Q n · . . . · Q 1 Ψ( x, γ ) = ( λ ( γ ) − l n ) . . . ( λ ( γ ) − l 1 )Ψ( x, γ ) (10) Lemma 4. L e t: f ( n ) ( x ) = Q n · Q n − 1 · . . . · Q 1 f ( x ) , wher e f ∈ F X . Then f ( n ) ( x ) is a c omplex smo oth p erio dic function. The pro of is straightforw ard: Each op erator Q k reduces the order n j of singularit y a t the p oint x j b y 1. Lemma 5. Assume, that the function f ( n ) ( x ) admits the L aur en t-F ourier de c om p osition in the Blo ch functions for the op er ator L n : f ( n ) ( x ) = X j c j ˜ Ψ ( n ) ( κ j , x ) , Then the function M − 1 f ( n ) ( x ) is wel l-define d, and we have M − 1 f ( n ) ( x ) = X j c j ( λ ( κ j ) − l 1 ) . . . ( λ ( κ j ) − l n )) ˜ Ψ ( n ) ( κ j , x ) . F o r al l κ j we have Im p ( κ j ) = 0 . Ther ef o r e we have no zer o e s in the d e n om- inators. W e also hav e: f ( n ) ( x ) = M ( M − 1 f ( n ) ( x )) = Q n · . . . · Q 1 · Q ∗ 1 · . . . · Q ∗ n · ( M − 1 f ( n ) ( x )) , and therefore f ( x ) = Q ∗ 1 · . . . · Q ∗ n · ( M − 1 f ( n ) ( x )) . (11) The results of this section can b e summarized in the following wa y . T o decomp ose any giv en function f ( x ) w e p erform fo llo wing actions: 1. By applying n prop erly chosen Da rb oux-Crum transformations w e ob- tain a smo oth functions f ( n ) ( x ) = Q n · Q n − 1 · . . . · Q 1 f ( x ). 15 2. W e decomp o se the smo oth function f ( n ) ( x ) in the eigenfunctions of the smo oth complex finite-gap op erato r L n . 3. T aking into a ccount, that all p oin ts q n are lo cated outside of the con- tour Im( p ) = 0, w e construct the eigenfunctions dec omp osition for M − 1 f n ( x ). 4. Applying form ula (11) we o btain an eigenfunction decomp o sition of the original function f ( x ). At this stage we use the follo wing prop erty of the Darb oux-Crum tra nsfor ma t io n: t he op erato r Q ∗ 1 · . . . · Q ∗ n · ( M − 1 f ( n ) ( x )) maps the Blo c h eigenfunctions of L n to the eigenfunctions of L . T o complete the pro of, it is sufficien t to prov e decomp ositions theorem for smo oth complex finite-gap o p erators. 1.3 Decomp osition for smo oth complex p erio dic p o- ten tials Remark. Recen t ly I.M. Kric hev er p oin ted out to the authors that the pro of of eigenfunctions decomp osition theorem for regula r complex finite- gap p oten tials was pro v ed in his pap er [4]. Consider a hyperelliptic Riemann surface Γ with divisor D . W e assume, that the corr espo nding p oten tial u ( x ) is regular and p erio dic with the perio d T , but ma y b e complex. It is natural to use the lo cal co ordinate 1 / k near infinit y where k = p ( γ ), λ = k 2 + O (1). In the neigh b ourho o d of infinit y w e ha v e: Ψ( x, γ ) = e ik x  1 + φ 1 ( x ) k + O  1 k 2  , Ψ ∗ ( y , γ ) = e − ik y  1 − φ 1 ( y ) k + O  1 k 2  , φ 1 ( x + T ) = φ 1 ( x ) . Let us denote Ξ( x, y , γ ) = Ψ( x, γ )Ψ ∗ ( y , γ ) ( λ ( γ ) − λ 1 ) . . . ( λ ( γ ) − λ g ) 2 p ( λ ( γ ) − E 0 ) . . . ( λ ( γ ) − E 2 g )) dλ ( γ ) There exists a constant K 0 suc h that (1) In the domain | k | > K 0 the function 1 /k is a w ell-defined lo cal co o r- dinate. 16 (2) The function Ψ( x, γ ) e − ik x is holomorphic in the v ariable 1 /k for x in the domain | k | > K 0 , | Im x | < ǫ . F or sufficien tly large k w e hav e Ξ( x, y , γ ) = e ik ( x − y )  1 + φ 1 ( x ) − φ 1 ( y ) k + χ 2 ( k , x, y ) k 2  dk where χ 2 ( x, y , k ) is holomorphic in 1 /k , x , y in the domain | k | > K 0 , | Im x | < ǫ , | Im y | < ǫ . Our purp ose is to study the conv ergence of the F ourier transformation. Let f ( x ) b e either a Sc h w artz class function or a Blo ch-perio dic function. 1. Case 1. The in tegral F ourier transform. Let f ( x ) b e a function with a finite supp ort. W e define: ˆ f ( γ ) = 1 2 π ∞ Z −∞ Ψ ∗ ( y , γ ) f ( y ) dy ( ˆ S ( K ) f )( x ) = I Im p ( γ )=0 , | Re p ( γ ) |≤ K ˆ f ( γ )Ψ( x, γ ) ( λ ( γ ) − λ 1 ) . . . ( λ ( γ ) − λ g ) 2 p ( λ ( γ ) − E 0 ) . . . ( λ ( γ ) − E 2 g )) dλ ( γ ) (12) Our purp ose is to sho w, that ( ˆ S ( K ) f ) ( x ) conv erges to f ( x ) a s K → ∞ . It is easy to to see that ( ˆ S ( K ) f ) ( x ) = 1 2 π ∞ Z −∞ S ( K , x, y ) f ( y ) dy where S ( K ; x, y ) = I Im p ( γ )=0 , | Re p ( γ ) |≤ K Ξ( x, y , γ ) (13) In this section we shall pro v e the following theorem: Prop ositions 1. L et us assume that: 17 (a) The sp e ctr al curve Γ is r e gular (i.e. has no multiple p oints), (b) The c ontour Im p ( γ ) = 0 is r e gular, i.e. d p ( γ ) 6 = 0 eve rywher e at this c ontour. Then: (a) The kernel S ( K , x, y ) has the fol low i n g structur e S ( K , x, y ) = S classic al ( K , x, y ) + S c orr e ction ( K , x, y ) wher e S classic al ( K , x, y ) = 2 sin( K ( x − y )) x − y is the c o rr esp ond i n g kern e l for the “standar d” inte gr al F ourier tr ansform, a n d S c orr e ction ( K , x, y ) uniformly c onver ges at any c omp a ct set in the ( x, y ) -p lane to a c ontinuous function S c orr e ction ( ∞ , x, y ) . (b) L et x do es not b elong to the supp ort of f ( y ) . Then ( ˆ S ( K ) f )( x ) → 0 for K → ∞ , and S c orr e ction ( ∞ , x, y ) ≡ 0 . Mor e over ( ˆ S ( K ) f ) ( x ) → 0 faster than an y de gr e e of K . 2. Case 2. The discrete F ourier transform. Let f ( x ) b e Blo c h- p erio dic with the p erio d T : f ( x + T ) = κ 0 f ( x ) , where κ 0 = e iT ϕ 0 is an unitary m ultiplier | κ 0 | = 1. Consider the set of all p oints κ j suc h, that e iT p ( κ j ) = κ 0 . Let us define ˆ f ( κ j ) = 1 T T Z 0 Ψ ∗ ( κ j , y ) f ( y ) d y The m ultipliers in the in tegrand ha v e opp o site Blo c h m ultipliers, there- fore w e can in tegrate ov er a n y basic p erio d. Let us define ( ˆ S ( N ) f )( x ) = X | ( p ( κ j ) − ϕ 0 ) T |≤ 2 π N ˆ f ( κ j )Ψ( κ j , x ) ( λ ( κ j ) − λ 1 ) . . . ( λ ( κ j ) − λ g ) 2 p ( λ ( κ j ) − E 0 ) . . . ( λ ( κ j ) − E 2 g ))  dλ ( γ ) dp ( γ )      λ = κ j W e hav e ( ˆ S ( N ) f )( x ) = T Z 0 S ( N , x, y ) f ( y ) dy 18 where S ( N , x, y ) = 1 T X | ( p ( κ j ) − ϕ 0 ) T |≤ 2 π N Ξ( κ j , x, y ) dp ( κ j ) (14) Prop ositions 2. Assume, that al l p oints κ j such that e iT p ( κ j ) = κ 0 ,ar e r e gular: (a) They do not c oincide with the multiple p oints (if they exists). (b) dp ( κ j ) 6 = 0 for al l j . Then (a) The kernel S ( N , x, y ) has the fol lowing structur e S ( N , x, y ) = S classic al ( N , x, y ) + S c orr e ction ( N , x, y ) wher e S classic al ( N , x, y ) = e iφ 0 ( x − y ) T sin  π (2 N +1) T ( x − y )  sin  π T ( x − y )  is the c orr esp on ding kernel for the “standar d” discr ete F ourier tr ansform and S c orr e ction ( N , x, y ) uniformly c onv e r ges i n the ( x, y ) - plane to the c ontinuous function S c orr e ction ( ∞ , x, y ) . (b) L et a p oint x do es no t b elon g to the supp ort of f ( y ) . T hen ( ˆ S ( N ) f )( x ) → 0 for N → ∞ , and S c orr e ction ( ∞ , x, y ) ≡ 0 . W e prov e now the first part of Prop osition 1. Let S ( K, x, y ) be the k ernel defined b y for m ula (13) W e assume , that the orien tat io n on this con tour is defined b y Re dp ( γ ) > 0. Let us fix a sufficien tly large constan t K 0 . Then w e can write S ( K ; x, y ) = I 1 ( x, y ) + I 2 ( K , x, y ) + I 3 ( K , x, y ) + I 4 ( K , x, y ) I 1 ( x, y ) = I Im p ( γ )=0 , | Re p ( γ ) |≤ K 0 Ξ( x, y , γ ) 19 I 2 ( K , x, y ) =   − K 0 Z − K + K Z K 0   e ik ( x − y ) dk I 3 ( K , x, y ) =   − K 0 Z − K + K Z K 0   e ik ( x − y )  φ 1 ( x ) − φ 1 ( y ) k  dk I 4 ( K , x, y ) =   − K 0 Z − K + K Z K 0   e ik ( x − y ) χ 2 ( k , x, y ) k 2 dk A standard calculation implies: I 2 ( K , x, y ) = 2 sin( K ( x − y )) x − y − 2 sin( K 0 ( x − y )) x − y Let us denote: S classical ( K , x, y ) = I 2 ( K , x, y ) + 2 sin( K 0 ( x − y )) x − y S correction ( K , x, y ) = I 1 ( x, y ) + I 3 ( K , x, y ) + I 4 ( K , x, y ) − 2 sin( K 0 ( x − y )) x − y The functions I 1 ( x, y ) and − 2 s in( K 0 ( x − y )) x − y do not depend on K and are con tin uous in b oth v ar ia bles. In tegral I 4 ( K , x, y ) absolutely con verges a s K → ∞ , therefore the limiting function is con tin uous in x , y . W e a lso ha v e I 3 ( K , x, y ) = 2 i Si ( K ( x − y ))( φ 1 ( x ) − φ 1 ( y )) − 2 i Si ( K 0 ( x − y ))( φ 1 ( x ) − φ 1 ( y )) , where Si ( x ) = x Z 0 sin( t ) t dt, therefore it unifo r mly con v erges to a con tinuous function I 3 ( ∞ , x, y ) = π i sign( x − y ))( φ 1 ( x ) − φ 1 ( y )) − 2 i Si ( K 0 ( x − y ))( φ 1 ( x ) − φ 1 ( y )) , This completes the pro of. 20 Corollary 2. The k e rnel S ( ∞ , x, y ) = lim K →∞ S ( K , x, y ) is a wel l- d efine d dis- tribution and S ( ∞ , x, y ) = 2 π δ ( x − y ) + S c orr e ction ( ∞ , x, y ) Pro of of the first part of Prop osition 2. Consider a sufficien tly larg e N 0 . It is natural to write S ( N , x, y ) = I 1 ( x, y ) + I 2 ( N , x, y ) + I 3 ( N , x, y ) + I 4 ( N , x, y ) where I 1 ( x, y ) = 1 T X | ( p ( κ j ) − ϕ 0 ) T |≤ 2 π N 0 Ξ( κ j , x, y ) dp ( κ j ) , I 2 ( N , x, y ) = 1 T " − 1 − N 0 X j = − N + N X j = N 0 +1 # e ( 2 πi T N + iϕ 0 ) ( x − y ) I 3 ( N , x, y ) = 1 T " − 1 − N 0 X j = − N + N X j = N 0 +1 # e ( 2 πi T j + iϕ 0 ) ( x − y ) 2 π T j + ϕ 0 ( φ 1 ( x ) − φ 1 ( y )) I 4 ( N , x, y ) = 1 T " − 1 − N 0 X j = − N + N X j = N 0 +1 # e ( 2 πi T j + iϕ 0 ) ( x − y ) χ 2 (2 π T j + ϕ 0 , x, y ) ( 2 π T j + ϕ 0 ) 2 A standard calculation implies: I 2 ( N , x, y ) = e iϕ 0 ( x − y ) T sin  π (2 N +1) x T  sin  π x T  − e iϕ 0 ( x − y ) T sin  π (2 N 0 +1) x T  sin  π x T  Let us denote: S classical ( N , x, y ) = I 2 ( N , x, y )+ e iϕ 0 ( x − y ) T sin  π (2 N 0 +1) x T  sin  π x T  = e iϕ 0 ( x − y ) T sin  π (2 N +1) x T  sin  π x T  S correction ( N , x, y ) = I 1 ( x, y )+ I 3 ( N , x, y )+ I 4 ( N , x, y ) − e iϕ 0 ( x − y ) T sin  π (2 N 0 +1) x T  sin  π x T  The term I 1 ( x, y ) is contin uous in x , y , I 4 ( N , x, y ) uniformly con v erges t o a con tin uous function. 21 The term I 3 ( N , x, y ) requires some extra att ention. It can b e written as: I 3 ( N , x, y ) = 1 2 π e iϕ 0 ( x − y ) " 1 − N 0 X j = − N + N X j = N 0 +1 # e ( 2 πi T j ) ( x − y ) j ( φ 1 ( x ) − φ 1 ( y ))+ − ϕ 0 T e iϕ 0 ( x − y ) " 1 − N 0 X j = − N + N X j = N 0 +1 # e ( 2 πi T j ) ( x − y ) ( 2 π T j + ϕ 0 )( 2 π T j ) ( φ 1 ( x ) − φ 1 ( y )) The second term unifor mly conv erges to a con tinuous function in x , y . Let us denote S 1 ( N , z ) = " 1 X k = − N + N X k =1 # e ik z k = i z Z 0 " sin  N + 1 2  w  sin  w 2  − 1 # dw F unction S 1 ( N , z ) is p erio dic with p erio d 2 π and con v erges to i ( π sign( z ) − z ) at the in t erv al [ − π , π ] uniformly outside any neigh b o rho o d of the p oin t z = 0. W e hav e I 3 ( N , x, y ) = 1 2 π e iϕ 0 ( x − y ) S 1 ( N , 2 π T ( x − y ))( φ 1 ( x ) − φ 1 ( y )) + regular terms therefore it a lso conv erges uniformly to a contin uous k ernel. Corollary 3. The kernel S ( ∞ , x, y ) = lim N →∞ S ( N , x, y ) is a w el l-define d dis- tribution and S ( ∞ , x, y ) = X j δ ( x − y − j T ) + S c orr e ction ( ∞ , x, y ) T o contin ue the pro o f w e need the follo wing: Lemma 6. L e t f ( y ) b e a smo oth function with c omp ac t supp ort. ˆ f ( γ ) = 1 2 π ∞ Z −∞ Ψ ∗ ( y , γ ) f ( y ) dy Then for any n ther e exists a c o n stant C n = C n ( u [ y ] , f [ y ]) such, that for sufficiently lar ge λ ( γ ) we have | ˆ f ( γ ) | ≤ C n | λ ( γ ) | n max y ∈ supp f ( y ) | e − ip ( γ ) y | 22 Pro of. By definition, we ha v e L n Ψ ∗ ( x, γ ) = λ ( γ ) n Ψ ∗ ( x, γ ). Therefore λ ( γ ) n ∞ Z −∞ Ψ ∗ ( y , γ ) f ( y ) dy = ∞ Z −∞ [ L n Ψ ∗ ( y , γ )] f ( y ) dy . F unction f ( y ) has a finite suppo r t, therefore w e can eliminate all deriv ativ es of Ψ ∗ ( y , γ ) b y in tegrating by parts, and w e obtain λ ( γ ) n ∞ Z −∞ Ψ ∗ ( y , γ ) f ( y ) dy = = ∞ Z −∞ Ψ ∗ ( y , γ ) P n ( f ( y ) , f ′ ( y ) , . . . , f (2 n ) ( y ) , u ( y ) , u ′ ( y ) , . . . , u (2 n − 2) ( y )) dy . where P ( . . . ) is a p olynomial. The function e ip ( γ ) y Ψ ∗ ( y , γ ) is uniformly b o unded for all y and sufficien tly large λ ( γ ) , therefore w e obtain the desire d estimate. Let us prov e the second par t of Prop osition 1. Consider the in tegral represen tation (12) for ( ˆ S ( K ) f ) ( x ). The integrand is holomorphic in γ at the finite part of Γ. F r om Lemma 6 w e see that the integral (1 2) absolutely con v erges as K → ∞ in the upp er half- plane if x > supp f ( y ) or in the lo w er half-plane if x < supp f ( y ). Moreov er the in tegrand exp onentially decreases in the corresp onding half-plane, therefore it is equal to 0. The inte grand deca ys at infinit y faster then any degree o f K , therefore the in tegral is fast deca ying as K → ∞ . In order to finish the pro o f of the second part of Prop osition 2, w e shall use the following in tegra l represen tation for S ( N , x, y ) S ( N , x, y ) = 1 2 π I β N Ξ( κ j , x, y ) e ip ( κ j ) T − κ 0 Here β N denotes the follo wing con tour (w e assume N to b e sufficie ntly large and p ( γ ) is fixed near infinity as a single-v a lued function): β N = β (1) N ∪ β (2) N ∪ β (3) N ∪ β (4) N β (1) N = { Im p ( γ ) = − N , | (Re p ( κ j ) − ϕ 0 ) T | ≤ 2 π ( N + 1 / 2) } , 23 β (2) N = {| Im p ( γ ) | ≤ N , (Re p ( κ j ) − ϕ 0 ) T = 2 π ( N + 1 / 2) } , β (3) N = { Im p ( γ ) = N , | (Re p ( κ j ) − ϕ 0 ) T | ≤ 2 π N } , β (4) N = {| Im p ( γ ) | ≤ N , (Re p ( κ j ) − ϕ 0 ) T = − 2 π ( N + 1 / 2 ) } , W e choose t he orien tation on β N b y assuming, that infinite p oin t is lo cated outside of the contour. By calculating the residues w e immediately obtain formula (1 4). Remark 2. At al l multiple p oints (if they exist) we have Im p ( E j ) = 0 . Ther efor e al l of them ar e inside the c ontour β N . F or a ny holomorp hic dif - fer ential on singular curve, the sum of r e s idues at sin gular p oi n ts is e qual to zer o. Ther efor e they do not affe ct our inte g r al. W e hav e ( ˆ S ( N ) f )( x ) = 1 2 π I β N x 0 + T Z x 0 Ξ( κ j , x, y ) e ip ( κ j ) T − κ 0 f ( y ) dy Let us c ho ose x = x 0 . The supp ort of f ( y ) do es not contain x , therefore w e can write ( ˆ S ( N ) f )( x ) = 1 2 π I β N x + T − ε Z x + ε Ξ( κ j , x, y ) e ip ( κ j ) T − κ 0 f ( y ) dy for some ε > 0. F rom Lemma 6 it f o llo ws, that for any M there exist constan ts D M suc h, that       x + T − ε Z x + ε Ξ( κ j , x, y ) e ip ( κ j ) T − κ 0 f ( y ) dy       ≤ D M e − N ǫT on β 1 , β 3       x + T − ε Z x + ε Ξ( κ j , x, y ) e ip ( κ j ) T − κ 0 f ( y ) dy       ≤ D M N M on β 2 , β 4 Therefore ( ˆ S ( N ) f )( x ) tends to 0 faster than an y degree of N as N → ∞ . 24 1.4 The reconstruction form ula for singular p oten tials T o complete the pro of, let us c hec k form ula (7). Let us denote η ( γ ) = ˜ Ψ( x, γ ) Ψ( x, γ ) . Then ˆ f ( γ ) = 1 2 π η ( σ γ ) ∞ Z −∞ ˜ Ψ( y , σ γ ) f ( y ) d y = = 1 2 π η ( σ γ ) ∞ Z −∞ ( Q ∗ 1 · . . . · Q ∗ n · Q n · . . . · Q 1 · ˜ Ψ( y , σ γ )) f ( y ) ( λ ( γ ) − l 1 ) . . . ( λ ( γ ) − l n ) dy = = 1 2 π η ( σ γ ) ∞ Z −∞ ( Q n · . . . · Q 1 · ˜ Ψ( y , σ γ ))( Q n · . . . · Q 1 · f ( y )) ( λ ( γ ) − l 1 ) . . . ( λ ( γ ) − l n ) dy = = 1 2 π η ( σ γ ) ∞ Z −∞ ˜ Ψ ( n ) ( y , σ γ ) f ( n ) ( y ) dy = 1 η ( σ γ ) ˆ f ( n ) ( γ ) . W e hav e f ( n ) ( x ) = I Im p ( γ )=0 ˆ f ( n ) ( γ ) ˜ Ψ ( n ) ( x, γ ) ( λ − λ 1 ( x 0 )) . . . ( λ − λ g − n ( x 0 ))( λ − l 1 ) . . . ( λ − l n ) 2 p ( λ − E 0 ) . . . ( λ − E 2 g )) dλ, where λ = λ ( γ ). f ( x ) = Q ∗ 1 · . . . · Q ∗ n · ( M − 1 f ( n ) ( x )) = = I Im p ( γ )=0 ˆ f ( n ) ( γ ) ˜ Ψ( x, γ ) ( λ − λ 1 ( x 0 )) . . . ( λ − λ g − n ( x 0 )) 2 p ( λ − E 0 ) . . . ( λ − E 2 g )) dλ = = I Im p ( γ )=0 η ( σ γ ) η ( γ ) ˆ f ( γ )Ψ( x, γ ) ( λ − λ 1 ( x 0 )) . . . ( λ − λ g − n ( x 0 )) 2 p ( λ − E 0 ) . . . ( λ − E 2 g )) dλ = T aking into accoun t, that η ( σ γ ) η ( γ ) = ( λ ( γ ) − λ 1 ) . . . ( λ ( γ ) − λ g ) ( λ − λ 1 ( x 0 )) . . . ( λ − λ g − n ( x 0 )) , w e obtain t he form ula (7). 25 App endix 2. Pro of of Statement 3 In this App endix w e presen t a pro of of the Statemen t 3. T o b e precise, w e pro v e the f o llo wing: Theorem. Assume, that we have the sp ac e F X asso ciate d with either a de - c aying at infinity p otential with N si ngular p oints of or ders n 1 ,. . . , n N or a p erio di c p otentials with N singular p oints o f or ders n 1 ,. . . , n N at the p erio d. In the p erio dic c ase we ass ume that an unitary Blo ch multiplier κ 0 , | κ 0 | = 1 is fixe d. 1. Denote by l X the fol lowing numb er l X =  n 1 +1 2  +  n 2 +1 2  + . . . +  n N +1 2  , wher e [ ] is the inte ger p art of a numb er. Ther e exists a n l X -dimensional subsp ac e of F X such that our sc alar pr o duct is n e gative define d o n it. 2. Any s ubs p ac e o f dimen sion d > l x has no n-zer o interse ction with F 0 X , i.e. c ontains at le ast one function w ith p ositive squar e. The pro of of the second part is straigh tforw ard. A function fr o m the space F X lies in F 0 X if it satisfies exactly l x linear equations: all singular terms in the expansions near p oin ts x j are equal t o 0. W e ha v e d - dimensional subspace with d > l X , t herefore t his system of equations has a least one non- t r ivial solution. T o prov e t he first part of the Theorem we construct these negative sub- spaces explicitly . In is con v enien t to consider t he deca ying and the p erio dic cases separately . 2.1 Deca ying at infinit y case. Let us prov e three tec hnical lemmas. Lemma 7. Assume, that we have o nly one singular p oint x 1 = 0 of or der n . F o r a n y n the f unctions 1 /x n − 2 l , l = 0 , 1 , . . . ,  n − 1 2  gener ate a zer o subsp ac e with r esp e ct to our sc alar pr o d uct: < 1 x n − 2 k , 1 x n − 2 l > = + ∞ Z −∞ dx x 2 n − 2 k − 2 l = 0 . 26 Here w e use o ur standard rule that the integration con tour go es around zero in the complex domain, 2 k < n , 2 l < n . The pro of is ob vious. Lemma 8. L et us assume, that we h ave on ly one singular p o i n t x 1 = 0 of the or der n , and N is an inte ger such, that N > n . Consider the fol lowin g c ol le ction of functions Ξ l ( x, ε ) , l = 0 , 1 , . . . ,  n − 1 2  in our sp ac e F X : Ξ l ( x, ε ) = 1 √ ε h ε x i n − 2 l e − [ x/ε ] 2 N , The Gr am m a trix g n k l for this c ol le ction of functions g n k l = < Ξ k ( x, ε ) , Ξ l ( x, ε ) > (15) is ne gative define d and do es not dep end on ε . Pro of. Consider an y linear com bination of these functions f ( x ) = n − 1 X k =0 d k Ξ k ( x, ε ) W e hav e < f , f > = 1 ε + ∞ Z −∞ n − 1 X k =0 n − 1 X l =0 d k d l h ε x i 2 n − 2 k − 2 l [ e − 2[ x/ε ] 2 N − 1] dx + + 1 ε + ∞ Z −∞ n − 1 X k =0 n − 1 X l =0 d k d l h ε x i 2 n − 2 k − 2 l dx The second in tegral is equal to zero by Lemma 7 and the in tegrand in the first in tegral is real, regular a nd strictly nega t ive, therefore < f , f > < 0 . The second statemen t immediately follo ws from the scaling prop erties. Lemma 9. We assume (as in the pr ev i o us le m ma) that we have only one singular p oin t at the p oint x 1 = 0 of the or der n , and N is an inte ger such 27 that N > n . L et us fix an interval [ − L, L ] wher e L i s either any p ositive numb er or + ∞ . Consider the fol lowin g c ol le ction of functions Ξ l ( x, ε ) , l = 0 , 1 , . . . ,  n − 1 2  in our sp ac e F X : Ξ l ( x, ε ) = 1 √ ε h ε x i n − 2 l · e − [ x/ε ] 2 N · ζ ( x ) , wher e ζ (0) 6 = 0 , ζ ( x ) is b ounde d on the interval [ − L, L ] and smo oth i n side it, ζ ′ (0) = ζ ′′ (0) = . . . = ζ (2 N − 1) (0) = 0 . Define the Gr am matrix ˜ g n k l ( ε ) for this c ol le ction of functions by ˜ g n k l ( ε ) = + L Z − L Ξ k ( x, ε ) Ξ l ( ¯ x, ε ) dx. (16) Then ˜ g n k l ( ε ) → g n k l · | ζ 2 (0) | as ε → 0 , wher e g n k l ar e the sc alar pr o ducts fr om L emm a 8. Her e we use our standa r d rule, that the inte gr ation c ontour go es ar ound the singular p oint x = 0 in the c omplex domain. Pro of. Let us mak e the following substitution: x = εy . W e hav e ˜ g n k l ( ε ) = + L/ε Z − L/ε  1 y  2 n − 2 k − 2 l · e − 2 y 2 N · ζ ( εy ) ζ ( ε ¯ y ) dy = = ζ (0) ζ (0) + L/ε Z − L/ε  1 y  2 n − 2 k − 2 l · e − 2 y 2 N dy + + + L/ε Z − L/ε  1 y  2 n − 2 k − 2 l · h ζ ( εy ) ζ ( ε ¯ y ) − ζ (0) ζ ( 0) i · e − 2 y 2 N dy The first in tegral con ve rges to g n k l | ζ 2 (0) | as ε → 0. The pre-exp onen t in the second integral is b ounded. It uniformly con v erges to 0 at any compact in terv al. Therefore this integral con v erges to 0. The pro of is finished. 28 Consider no w the generic ”deca ying at infinity” case. W e assume that o ur p oten tial has N singular p oints x 1 ,. . . , x N with the m ultiplicities n 1 ,. . . , n N . Let N = max( n 1 , . . . , n N ) + 1. Consider the fo llowing collection of f unctions Ξ lj ( x, ε ) = 1 √ ε ·  ε x − x j  n j − 2 l e − h x − x j ε i 2 N · ζ j ( x ) , where ζ j ( x ) =    Y k 6 = j k =1 ,...,N (( x − x j ) 2 N − ( x k − x j ) 2 N ) 2 (( x − x j ) 2 N − ( x k − x j ) 2 N ) 2 + 1    N l = 0 , . . . , h n j − 1 2 i , j = 1 , . . . , N . Lemma 10. Al l functions Ξ lj b elong to the sp ac e F X . Pro of. The function Ξ lj ( x, ε ) are symmetric in ( x − x j ) if n j is ev en or sk ew-symmetric in ( x − x j ) if n j is o dd. A t all other singular p oints x l they ha ve zero es of order 2 N ≥ n l + 1. At infinit y they deca y exp onen tially , therefore all conditions are fulfilled. Lemma 11. The sc alar pr o ducts of functions define d ab ove have the fol lowing form < Ξ l 1 j 1 ( x, ε )Ξ l 2 j 2 ( x, ε ) > = g n j 1 l 1 l 2 · ζ 2 j 1 (0) · δ j 1 j 2 + O (1) as ε → 0 . (17) wher e g n k l denotes the Gr am matrix define d by (15). Pro of. Let j 1 6 = j 2 . Assume that x j 1 < x j 2 , 2 L = x j 2 − x j 1 . The pro duct Ξ l 1 j 1 ( x, ε )Ξ l 2 j 2 ( x, ε ) is regular in t he whole x -line. Consider the follo wing system of in terv als in the x -line. I 1 =] −∞ , x j 1 − L ] , I 2 = [ x j 1 − L, x j 1 + L ] , I 3 = [ x j 1 + L, x j 2 + L ] , I 4 = [ x j 2 + L, + ∞ [ . Then w e hav e the following estimates: | Ξ l 1 j 1 ( x, ε )Ξ l 2 j 2 ( x, ε ) | ≤ ε n j 1 + n j 2 − 2 l 1 − 2 l 2 − 1 L n j 1 + n j 2 − 2 l 1 − 2 l 2 e − 2[ L/ε ] 2 N for x ∈ I 1 ∪ I 4 . 29 | Ξ l 1 j 1 ( x, ε )Ξ l 2 j 2 ( x, ε ) | ≤ ε n j 1 + n j 2 − 2 l 1 − 2 l 2 − 1 [2 N ] 2 N 3 4 N 2 − 2 N L n j 1 − 2 l 1 − 4 N 2 e − [ L/ε ] 2 N for x ∈ I 2 | Ξ l 1 j 1 ( x, ε )Ξ l 2 j 2 ( x, ε ) | ≤ ε n j 1 + n j 2 − 2 l 1 − 2 l 2 − 1 [2 N ] 2 N 3 4 N 2 − 2 N L n j 2 − 2 l 2 − 4 N 2 e − [ L/ε ] 2 N for x ∈ I 3 F rom thes e estimates it follo ws that all in tegrals exponentially deca ys as ε → 0. F or j 1 = j 2 the asymptotic form ula for the scalar pro ducts immediately follo ws from Lemma 9. As a corollary of Lemma 11 w e immediately obtain, that for sufficien tly small ε > 0 t he scalar pro duct o n our system of functions is negativ e defined. It completes the pro of for f ast deca ying case. 2.2 P erio dic case. T o simplify the form ulas below w e assume tha t the p erio d of our p otential is equal to π in this section. Consider the fo llowing collection of f unctions Ξ lj ( x, ε ) = 1 √ ε ·  ε sin( x − x j )  n j − 2 l e −  sin( x − x j ) ε  2 N · ζ j ( x, ε ) · e ic j α ( x ) , where ζ j ( x, ε ) =    Y k 6 = j k =1 ,...,N ([sin( x − x j )] 2 N − [sin( x k − x j )] 2 N ) 2 ([sin( x − x j )] 2 N − [sin( x k − x j )] 2 N ) 2 + 1    N , α ( x ) = x R 0 N Q k =1 [sin( y − x k )] 2 N dy π R 0 N Q k =1 [sin( y − x k )] 2 N dy l = 0 , . . . , h n j − 1 2 i , j = 1 , . . . , N , the constan ts c j are c hosen to provide the prop er p erio dicit y: e ic j = ( − 1) n j κ 0 . It is easy to c hec k, that all functions Ξ lj b elong to the space F X . 30 Lemma 12. The sc alar pr o ducts of the functions define d ab ove have the fol lowing form < Ξ l 1 j 1 ( x, ε )Ξ l 2 j 2 ( x, ε ) > = g n j 1 l 1 l 2 · ζ 2 j 1 (0) · δ j 1 j 2 + O (1) as ε → 0 . (18) wher e g n k l denotes the Gr am matrix define d by (15). Pro of. Let j 1 6 = j 2 . Then all pro ducts Ξ l 1 j 1 ( x, ε ) Ξ l 2 j 2 ( ¯ x, ε ) are regular functions, uniformly exp onen tially conv erging to 0 as ε → 0. Let j 1 = j 2 . It is sufficien t to in tro duce a new v ariable y = sin ( x ). After that w e a pply Lemma 9 This completes the pro of of t he Statemen t 3. App endix 3 Let us consider the followin g Problem: Ho w many p o les the real singular T - p erio dic finite-ga p 1D Sc hro dinger Operat o r migh t hav e at the p erio d [0 , T ]? This question is esp ecially in teresting for the case of R-F ourier T ransform. W e presen t here a complete answ er to this question fo r the Riemann surfa ces Γ with real branching p oints w 2 = ( λ − E 0 ) ... ( λ − E 2 g ) , E k ∈ R F or the smo oth p erio dic op erator with the same sp ectral curv e Γ w e ha v e sp ectral zones [ E 0 , E 1 ] , [ E 2 , E 3 ] , ..., [ E 2 g − 2 , E 2 g − 1 ] , [ E 2 g , ∞ ] Consider the trace of mono drom y matrix T ( λ ) corresp onding to the p erio d T . It has k j maxima and minima strictly inside of the sp ectral zone [ E 2 j − 2 , E 2 j − 1 ] , j = 1 , . . . , g , k j ≥ 0 Theorem. L et the p otential u ( x ) c orr esp ond to the c as e of R- F ourier T r ans- form and R iemann surfac e Γ as ab ove. The total numb er of singularities at the p erio d [ x, x + T ] is e qual to the numb er n ( u ) = ( k g + 1) + ( k g − 2 + 1) + . . . + ( k g − 2 i + 1) + . . . + ( k 2 + 1) , g = 2 s 31 n ( u ) = ( k g + 1) + ( k g − 2 + 1) + . . . + ( k g − 2 i + 1) + . . . + ( k 1 + 1) , g = 2 s + 1 Each singulari ty has a lo c al form ∼ n l ( n l + 1) / ( x − x l ) 2 . F or x l = 0 we have singularity w ith n 0 = g . Ther e exists only one singular p oint with n l = g , i.e. n l < n 0 for l 6 = 0 . By de fi nition, n ( u ) = P l n l F or the fa mo us La m ´ e p otentials u = g ( g + 1) ℘ ( x ) corresp onding to the rectangular lattice w e ha v e all k j = 0. Our requiremen t ( n ( u ) = g ) implies that k j = 0 for the ev en v alues of j + g only . So there are man y R-F ourier T ransform p otentials of tha t type (i.e. with one p ole at the p erio d where t he Hermit t yp e Dirichle t Pro blem mak es sense). The pro of of t his theorem is based on the results of t his w ork: A comparison of the n um b er of negativ e squares in the inner pro duct with the n um b er of poles of p otential, and the description of the functional space F X are necessary for the pro o f. It is easy to classify all realizable collections k j . The case of Lam ´ e type elliptic p otentials corresponding to the rhom bic lattices, as w ell as more general cases o f Riemann surfaces with complex branching p oints will b e described in t he next w ork. Remark 3. a)The main go al of our wo rk is to develop the the ory of the R-F o urier T r ansform wh i c h is the b est p ossib le analo g of F ourier tr ansform (with go o d multiplic ative pr op erties) on Riemann surfac es. It sho uld b e ap- plie d to the sp ac es of smo oth functions define d in the sp e cial ”c anonic al”c ontours on the r e al Riemann surfac e s. It involves quite original sp e ctr al the ory for the sin g ular op er ators o n the whole r e al x -line. This sp e ctr al the ory i s b ase d on the sp e cial indefinite inner pr o ducts in the sp ac es of functions, c ontaining singular func tion s . We r e alize d this p r o gr am for the r e al singular finite-gap (i.e. algebr o- g e ometric al ) op er ators, b ut it c ertainly c an b e extend e d to the infinite-gap c as e s a lso. Th e c orr esp onding op er ator L is define d in the sam e sp ac e F X as b efor e. It is symm e tric in the same indefinite inn e r pr o duct. However we did not pr ove d yet the c ompletene s s of the c orr esp on ding b as i s . A n obstacle for that c an b e fo und in the works [9], [10], [11], [12]. The Darb oux–Crum tr ansformations play only te chnic al r ole her e. We do not c onsider them as a r e al ly ne c es s ary p art of our the ory. b) I n our work we inve n te d the fol lowin g class o f r e al p otentials with the sp e cia l isolate d singularities: for al l values of c om plex sp e ctr al p ar ameter the solutions ψ ( x ) should b e lo c al ly mer omorphic in the infinitesimal neighb or- ho o d of the c orr esp ond ing singular p oints at the r e al x -axis. We found no 32 classic al or mo dern works wher e this pr o p erty ha d b e en discusse d . The anal- o gous pr op erty has b e en use d in the work [1 3] for the solutions define d i n the whole c omplex x -pla n e but for the e l liptic p otentials only: It was use d as an assumption implying that such el liptic p otential is algeb r o-ge om etric al – i.e. “singular finite gap”. In our opinion, this is the most essential id e a of the work [13] r elate d to our the ory. c) Ther e is a pr oblem c onc erning a c ompletion of our functional c l a s ses. We pr ove d the de c omp os i tion the or em s fo r lo c a l ly smo oth functions ( o utside of singularities), but Hilb ert Sp ac es do not work her e. Remark 4. a)De c oninck and Se gur made the fol lo w ing claim i n the se ction 4.4 of their work [14]: A c c or ding to the KdV dynamics (with hier ar chy) the p oles of finite-gap el liptic, r ational and trigonom etric solution c an c ol lapse to singular p oints by the trian g ular gr oups c ontainin g n ( n + 1) / 2 items only. It is true, but they claim a lso that the p oles ar e le aving this p oint in the c ompl e x x -plane along the dir e ctions of vertic es of the e quilater al p olygon. It is wr ong for n > 2 . I n fact, ac c o r ding to our r esults [1], [2] this multiple p ole splits “appr oxima tely” ( b ut no t exactly) as a set of inte ger p o i n ts in the e quilater al triangle for al l n = 1 , 2 , 3 , . . . . Exactly [( n + 1) / 2] p oles r emain at the r e al ax i s which is a diagonal in this triangle . 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