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Laplace transformations and spectral theory of two-dimensional semidiscrete and discrete hyperbolic Schrödinger operators. (English) Zbl 1080.47033

It is known that the two-dimensional hyperbolic Schrödinger operator is related to the two-dimensional Toda lattice via a chain of Laplace transformations, and if the operator is periodic, there is an algebro-geometric spectral theory. The discrete version of such a theory is also known [S. P. Novikov and I. A. Dynnikov, Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds, Russ. Math. Surv. 52, 1057–1116 (1977; Zbl 0928.35107); translation from Usp. Mat. Nauk 52, No. 5, 175–234 (1997)]. In this paper, for the semidiscrete hyperbolic Schrödinger operator

(Lψ) n =a n (y)ψ n (y)+b n (y)ψ n ' (y)+c n (y)ψ n+1 (y)+d n (y) n+1 (y),(1)

b n 0, d n 0, such theory is given.

For this purpose, introducing the shift operator Tψ n (y)=ψ(y)ψ n+1 , the authors rewrite (1) as

L=f n (y)( y +A n (y)))1+ν n (y)T)+w n (y)),L=f ^ n (y)((1+ν ^ n (y)T)( y +A ^ n (y))+w ^ n (y))·

Then the first and the second type Laplace transformations

LL ˜=f n (w n (1+ν n T)1 w n (+A n )+w n ),ψψ ^=(1+ν n T)ψ,LL ^=f ^ n (w ^ n (+A ^ n )1 w ^ n (1+ν ^ n T)+w ^ n ),ψψ ^=(+A ^ n )ψ,

are defined (Lemma 2.1 and Def. 2.2).

Introducing gauge transformations LL ¯, ψψ ¯=g n -1 ψ, it is shown the Laplace transforms of the first and the second type are inverse to each other as transformations of gauge equivalence class (Lemma 2.5). By using a chain of Laplace transforms of the first type, a family of solutions of the semidiscrete 2D Toda lattice

(g n k -g n+1 k ) ' =e g n k+1 -g n+1 k -eg n k -g n+1 k-1 ,

parametrized by an arbitrary function g0 0 (y) and a set of arbitrary constant r k , k can be obtained. Its converse is also shown (Th. 2.7). Parallel results for the two-dimensional discrete hyperbolic Schrödinger operator

(Lψ) n,m =a n,m ψ n,m +b n,m ψ n+1,m +c n,m ψ n,m+1 +d n,m ψ n+1,m+1 ,

are also given (§ 2.2).

Then considering the periodic operator of b n =-1, d n =1, the algebro-geometric spectral theory of B. A. Dubrovin, I. M. Krichever and S. P. Novikov [The Schrödinger equation in a periodic field and Riemannian surfaces, Sov. Math., Dokl. 17 91976], 947–951 (1977; Zbl 0441.35021); translation from Dokl. Akad. Nauk SSSR 229, 15–18 (1976)] is extended to the semidiscrete operators. Since the semidiscrete Laplace transform is obtained, discussions are parallel to the case of periodic two-dimensional Schrödinger operators. The same theory for discrete operators was also given in [I. M. Krichever, Two-dimensional periodic difference operators and algebraic geometry, Sov. Math. Dokl. 32, 623–627 (1985; Zbl 0603.39004); translation from Dokl. Akad. Nauk SSSR 285, 31–36 (1985)(§ 3)].

In § 4, the last section, the action of the Laplace transform on spectral data is determined (Th.4.2). It provides a theta-function description of solutions of the semidiscrete Toda lattice. The same result for the discrete case was already given in [A. A. Oblomkov, Difference operators on two-dimensional regular lattices, Theor. Math. Phys. 127, No. 1, 435–445 (2001; Zbl 0998.39012); translation from Teor. Mat. Fiz. 127, No. 1, 34–46 (2001)], implicitly. In this paper, an explicit description of the statement of Oblomkov’s paper is given in § 4.2.

MSC:
47B39Difference operators (operator theory)
14H70Relationships of algebraic curves with integrable systems
35A22Transform methods (PDE)
35P05General topics in linear spectral theory of PDE
35Q53KdV-like (Korteweg-de Vries) equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K20Relations of infinite-dimensional systems with algebraic geometry, etc.