Periodic one-point rank one commuting difference operators

P erio dic one-p oin t rank o n e comm uting difference o p e r a tors Alina Dobrogo wsk a ∗ and Andrey E. Mirono v † Abstract In this paper w e s tudy one-p oin t rank one comm utative rings of difference op erators. W e find conditions on sp ectral data which sp ecify suc h op erators with p erio dic co efficien ts. Keyw ords: comm uting difference op erators 1 In tro duction and Main Results In t his pap er w e study one–p o in t rank one commu ting difference op erato rs with p erio dic co efficien ts. Let us consider a (maximal) comm utativ e ring A of difference o p erators consisting of op erators of the form L m = i = N + X i = − N − u i ( n ) T i , u N + = 1 , N + , N − ≥ 0 , where T is a shift op erator, T ψ ( n ) = ψ ( n + 1) , m = N + + N − is the o der of L m (assuming u N − 6 = 0 ). The op erator L m acts on the space of formal functions { ψ : Z − → C } . The ring A is isomorphic to a ring of ∗ A.D. w as supp orted by the Polish Ministry of Science and Higher Education under subsidy for maintaining the r esearch p otential o f the F ac ul ty of Mathematics and Infor- matics, Univ ersity of Bialysto k † A.E.M. w a s supp orted by RFBR (grant 18-01-004 11) and by the State Mainte- nance Progr am for the Leading Scientific Schools of the Russian F eder ation (grant NSh- 5913.2 018.1 ) 1 rational functions on s p ectral curv e Γ with p oles in p oin ts q 1 , . . . , q s ∈ Γ (see [1]). Common eigenfunctions of o p erators from A form a v ector bun- dle of rank l ov er Γ \{∪ s j =1 q j } . More precisely , there is a v ector-function ψ ( n, P ) = ( ψ 1 ( n, P ) , . . . , ψ l ( n, P )) , P ∈ Γ \{∪ s j =1 q j } which is called Baker– A k h iezer function , suc h t ha t ev ery op erator L m ∈ A corresp onds to a mero- morphic function f ( P ) on Γ with p oles in q 1 , . . . , q s L m ψ = f ψ . Moreo v er, m = l m ′ , where m ′ is the degree of the pole divisor o f f . The op erators from A are called s - p oint r ank l op er ators . T wo–point rank o ne op erato rs w ere classified in [1, 2 ]. Bak er–Akhiezer function of s uc h op erators can b e reconstructed from Kriche v er’s spectral data [1]. One-p o int rank l > 1 op erators we r e disco v ered by Kric hev er a nd No vik ov in [3 ]. S p ectral data for one-p oint rank one op erators w ere found in [4]. Those op erato rs contain the shift op erator T only in p ositiv e p o wer. Recall that the sp ectral data fo r such op erators has the form (see [4]) S = { Γ , γ , P n , q , k − 1 } . Here Γ is a Riemannian surface of gen us g (we do not consider singular sp ectral curv es), γ = γ 1 + . . . + γ g is a non-sp ecial divisor on Γ , P n ∈ Γ , n ∈ Z is a set o f general p oin ts, q ∈ Γ is a fixed p oint, k − 1 is a lo cal parameter near q . There is a unique Bak er–Akhezer function ψ ( n, P ) , n ∈ Z , P ∈ Γ whic h is rational function on Γ and satisfies the follow ing prop erties • if n > 0 , then the zero a nd p ole divisor of ψ has the form ( ψ ( n, P )) = γ ( n ) + P 0 + . . . + P n − 1 − γ − nq , • if n < 0 , then the zero a nd p ole divisor of ψ has the form ( ψ ( n, P )) = γ ( n ) − P − 1 − . . . − P n − γ − nq , • if n = 0 then ψ ( n, P ) = 1 , • in a neigh b orho o d of q the function ψ has the follo wing expansion ψ ( n, P ) = k n + O ( k n − 1 ) . 2 Here γ ( n ) = γ 1 ( n ) + . . . + γ g ( n ) , n 6 = 0 is some divisor on Γ . F urther w e will use the follo wing notation γ (0) = γ . F or a rbitrary meromorphic function f ( P ) on Γ with t he unique p ole in q of order m there is a unique op erator L m = T m + u m − 1 ( n ) T m − 1 + . . . + u 0 ( n ) suc h that L m ψ = f ψ , see [4]. If in the sp ectral data S all p oin ts P n coincide, P n = q + , then w e get the t wo-point K ric hev er’s construction [1]. One-p oin t rank one op erators w ere studied in [4, 6, 7], in particular some explicit examples of suc h op erat o rs w ere giv en. That class of o p erators is v ery in teresting b ecause, for example with the help of those op erators one can construct a discretization of the Lamé op erator preserving the sp ectral curv e. More precisely , let ℘ ( x ) , ζ ( x ) b e the W eierstrass functions. W e define the function A g ( x, ε ) b y the follo wing form ulas A 1 = − 2 ζ ( ε ) − ζ ( x − ε ) + ζ ( x + ε ) , A 2 = − 3 2 ( ζ ( ǫ ) + ζ (3 ε ) + ζ ( x − 2 ε ) − ζ ( x + 2 ε )) , A g = A 1 g 1 Y i =1  1 + ζ ( x − (2 i + 1) ε ) − ζ ( x + (2 i + 1) ε ) ζ ( ε ) + ζ ((4 i + 1) ε )  , for o dd g = 2 g 1 + 1 , A g = A 2 g 1 Y i =2  1 + ζ ( x − 2 iε ) − ζ ( x + 2 iε ) ζ ( ε ) + ζ ((4 i − 1) ε )  , for eve n g = 2 g 1 . The op erator L 2 = 1 ε 2 T 2 ε + A g ( x, ε ) 1 ε T ε + ℘ ( ε ) (1) comm utes with the operator L 2 g +1 , op erator s L 2 , L 2 g +1 are rank o ne one- p oin t op erators. I n the ab ov e formulas it is assumed that T ε ψ ( x ) = ψ ( x + ε ) . The op erator L 2 has the followi ng expansion L 2 = ∂ 2 x − g ( g + 1) ℘ ( x ) + O ( ε ) . F or small g it is c hec ke d that the sp ectral curve of the pair L 2 , L 2 g +1 coincides with the spectral curv e of the Lamé op erator ∂ 2 x − g ( g + 1) ℘ ( x ) , see [6]. Probably this class of difference op erators can b e used for the construction of a discretiz ation of arbitrary finite-gap one dimensional Sc hrö dinger op erators. Note that the op erator (1) is p erio dic. So, for the discretization of the finite- gap op erators it is useful to find the condition when ra nk one one-p oin t 3 op erators are p erio dic with real co efficien ts. This is the main motiv ation of this pap er. In the nex t theorem we form ulate perio dicity and reality conditions of the co efficien ts of the op erators. Theorem 1 Co efficie nts of one-p oint r ank one op er ators c orr esp onding t o the sp e ctr a l d a ta S = { Γ , γ , P n , q , k − 1 } ar e N –p erio dic, N ∈ N , if and only if P n + N = P n , and ther e is a mer omorphic function λ ( P ) on Γ with a divis o r of zer os and p oles of the form ( λ ( P )) = P 0 + . . . + P N − 1 − N q . L et us assume that the sp e ctr al curve Γ admits an antiholom orphic invo- lution τ : Γ − → Γ , τ 2 = id. If τ ( P n ) = P n , τ ( γ ) = γ , τ ( q ) = q , τ ( k ) = k , (2) then the Baker–Akhiezer function satisfies the identity ψ ( n, P ) = ψ ( n, τ ( P )) , (3) and if additional ly τ ( f ( P )) = f ( P ) , then the c o efficients of the op er ator L m c orr esp o nding to the function f ( P ) , L m ψ = f ψ , ar e r e al. In the case of t w o- p oin t rank one op erators the analogue of Theorem 1 w as prov ed in [5]. In the t wo-p o in t case w e hav e ( λ ) = N q + − N q . 4 2 Pro of of T heorem 1 In the b eginning we pro ve the second pa r t of the theorem. Th e pro of of this part is u sual. The iden tity (3 ) follo ws from the uniquene ss of the Ba ker– Akhiezer function with the fixed sp ectral data. Indeed, from (2) it follow s that t he function ψ ( n, τ ( P )) satisfies the same conditions as ψ ( n, P ) , hence w e get (3). W e ha ve L m ψ ( n, τ ( P )) = f ( τ ( P )) ψ ( n, τ ( P )) . Conseque n tly , L m ψ ( n, τ ( P )) = ¯ L m ψ ( n, τ ( P )) = ¯ L m ψ ( n, P ) = f ( τ ( P )) ψ ( n, τ ( P )) . Hence ¯ L m ψ ( n, P ) = f ( P ) ψ ( n, P ) . F rom the uniqueness of the op erator corresp onding to the meromorphic func- tion f ( P ) we get ¯ L m = L m , hence, the co efficien ts of L m are real. T o pro ve the first part of the theorem w e introduce the fo llo wing function χ ( n, P ) = ψ ( n + 1 , P ) ψ ( n, P ) . F rom the definition of the Bake r–Akhiezer function w e obtain that the zero and p ole divisor of χ ha s the form ( χ ( n, P )) = γ ( n + 1) + P n − γ ( n ) − q , n ∈ Z . (4) Lemma 1 Op er ators fr om A have N -p erio dic c o efficients if and only if χ ( n + N , P ) = χ ( n, P ) . Pr o o f of L emma 1. Let us pro ve the in v erse part of the lemma. W e assume that the co efficien ts of all op erators from A are perio dic. This means that the op erator T N comm utes with all op erato rs from A , i.e., T N ∈ A . This also means that there is a meromorphic function λ ( P ) on Γ with the unique p ole in q of order N suc h that T N ψ ( n, P ) = ψ ( n + N , P ) = λ ( P ) ψ ( n, P ) . (5) 5 W e ha ve χ ( n + N , P ) = ψ ( n + 1 + N , P ) ψ ( n + N , P ) = λ ( P ) ψ ( n + 1 , P ) λ ( P ) ψ ( n, P ) = χ ( n, P ) . (6) Let us prov e the direct part of the lemm a. W e assume that χ ( n + N , P ) = χ ( n, P ) . W e in tro duce a rational function on Γ λ ( P ) = χ (0 , P ) . . . χ ( N − 1 , P ) = ψ ( N , P ) . Then w e obtain T N ψ ( n, P ) = ψ ( n + N , P ) = χ ( n + N − 1 , P ) ψ ( n + N − 1 , P ) = χ ( n + N − 1 , P ) χ ( n + N − 2 , P ) ψ ( n + N − 2 , P ) = . . . = χ ( n + N − 1 , P ) . . . χ ( n, P ) ψ ( n, P ) = χ (0 , P ) . . . χ ( N − 1 , P ) ψ ( n, P ) . Hence, T N ψ ( n, P ) = λ ( P ) ψ ( n, P ) . (7) F rom (7) it follow s that T N ∈ A since T N and op erators from A hav e common Bak er–Akhiezer eigenfunction. Moreo v er λ ( P ) has the unique pole of order N i n q . Lem ma 1 is prov ed. No w w e can finish the pro of of Theorem 1. Let us assume that co efficien ts of the op erators are p erio dic. Then by Lemma 1 the function χ is p erio dic and from (4) w e ha ve ( χ ( n + N , P )) = γ ( n + N + 1) + P n + N − γ ( n + N ) − q . (8) Hence, comparing the p ole divis ors of (4) a nd (8) we g et γ ( n ) = γ ( n + N ) , and after comparing the zero divisors of ( 4 ) and (8) w e get P n + N = P n . F rom the pro o f of Lemma 1 it follows that the function λ ( P ) = ψ ( N , P ) has an unique p ole q of order N , moreov er ( λ ( P )) = γ ( N ) + P 0 + · · · + P N − 1 − γ (0) − N q = P 0 + · · · + P N − 1 − N q . Hence the direct part of Theorem 1 is pro ve n. Let us assume that there is a meromorphic function λ ( P ) suc h that ( λ ( P )) = P 0 + · · · + P N − 1 − N q , 6 and P n + N = P n . W e can supp o se that in the neigh b orho o d of q w e hav e the expansion λ = k N + O ( k n − 1 ) . Then from (4) w e ha ve ( χ ( n + N )) = γ ( n + N + 1) + P n + N − γ ( n + N ) − q = γ ( n + 1) + P n − γ ( n ) − q = χ ( n ) . Since χ ( n ) = k + O (1) in the neighborho o d of q , we get χ ( n + N ) = χ ( n ) . Hence by Lemma 1 the co efficien t of the op erators are p erio dic. Theorem 1 is pro ven . 2.1 Example Let us consider the case of elliptic sp ectral curv e Γ giv en b y the equation w 2 = F ( z ) = z 3 + c 2 z 2 + c 1 z + c 0 . The degree of the divisor γ ( n ) is 1. L et γ ( n ) = ( α n , β n ) ∈ Γ , β 2 n = F ( α n ) . Comm uting op erators of orders 2 and 3 ha v e the f orms (see [4]) L 2 = ( T + U n ) 2 + W n , U n = β n + β n +1 α n +1 − α n , W n = − c 2 − α n − α n +1 , L 3 = T 3 + ( U n + U n +1 + U n +2 ) T 2 + ( U 2 n + U 2 n +1 + U n U n +1 + W n − α n +2 ) T + ( U n ( U 2 n + W n − α n ) + β n ) . The function χ ( n, P ) has the form χ = w + β n z − α n + β n + β n +1 α n − α n +1 . The p oin t P n = ( z n , w n ) ∈ Γ has the co ordinates z n = c 1 ( α n + α n +1 ) + α n α n +1 ( α n + α n +1 ) + 2 c 2 α n α n +1 + 2( c 0 + β n β n +1 ) ( α n − α n +1 ) 2 , 7 w n = β n +1 ( α n − z n ) + β n ( α n +1 − z n ) α n − α n +1 . If α n + N = α n , β n + N = β n , then γ ( n + N ) = ( α n + N , β n + N ) = ( α n , β n ) = γ ( n ) , P n + N = P n , and the meromorphic function λ ( P ) = χ (0 , P ) . . . χ ( N − 1 , P ) satisfies the conditions of Theorem 1. References [1] I.M. Kric hev er, Algebr aic curves a n d non–line ar differ enc e e qua- tions , Russian Math. Surv eys, 33 (4), 215-216, 1978. [2] D. Mumford, An algebr o–ge ome tric c onstruction of c ommuting op- er ators and solutions to the T o da la ttic e e quation, kortewe g–de V ries e quation and r elate d non- l i n e ar e quations , Pro ceedings of the Inte r - national Symposium on Algebraic Geometry , Kyoto, Japan, 1 9 77, Kinokuniy a, T oky o, 115-153, 1978. [3] I.M. Kric hev er, S.P . Novik o v, Two–dimen sional T o da lattic e, c om- muting differ enc e op er ators, and h o lomorphic bund les , Russian Math. Surv eys, 58 (3), 473-510 , 2003. [4] G.S. Maulesho v a, A.E. Mirono v, One–p oint c ommuting differ enc e op er ators of r ank 1 , Doklady Mathematics, 93 (1), 62-64, 2016. [5] I.M. K r ichev er, Commuting D iffer enc e Op er ators and the Combi- natorial Gale T r ansform , F unctional Analysis and Its Applications, 49 :3, 2015, 175–18 8 . [6] G.S. Maulesho v a , A.E. Mirono v, One-p oint c ommuting differ enc e op er ators of r ank 1 and t heir r e lation with fi nite-gap Schr o dinge r op er ators. Doklady Mathematics, 97 :1, 2018, 62–64 . [7] G.S. Maulesho v a, A.E. Mirono v, Positive one-p oint c ommuting d i f - fer enc e op er ators, Integrable Systems and Algebraic Geometry , V ol. 1, P . 383- 4 00 LMS Lecture Note Series (arXiv e:1810.107 17). 8 Alina Dobrogow sk a Institute of Mathematics, Univ ersity of Bialstok, Ciolo wskiego 1M, 15-245 Bialstok, P ola nd, E-mail: alina.dobrogo wsk a@u wb.edu.pl Andrey Mirono v Sob olev Institute of Mathematics, Nov osibirsk, Russia, and No vos ibirsk State Univ ersit y , No v osibirsk, Russia. E-mail: mirono v@math.nsc.ru 9