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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 [{\it S. P. Novikov} and {\it 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 $$(L\psi)_n = a_n(y)\psi_n(y) + b_n(y)\psi_n'(y) + c_n(y)\psi_{n+1}(y) + d_n (y)_{n+1}(y),\tag1$$ $b_n\ne 0$, $d_n\ne 0$, such theory is given. For this purpose, introducing the shift operator $T\psi_n(y) = \psi(y)\psi_{n+1}$, the authors rewrite (1) as \aligned & L = f_n(y) (\partial_y+A_n(y))) 1+\nu_n(y)T)+ w_n(y)),\\ & L = \widehat f_n(y) ((1 +\widehat\nu_n(y)T)(\partial_y+\widehat A_n(y))+\widehat w_n(y)).\endaligned Then the first and the second type Laplace transformations \align & L\mapsto\widetilde L= f_n(w_n(1+\nu_n T)\frac 1{w_n}(\partial+A_n)+w_n),\ \psi\mapsto \widehat\psi=(1+\nu_nT)\psi,\\ & L\mapsto\widehat L= \widehat f_n(\widehat w_n(\partial +\widehat A_n)\frac 1{\widehat w_n}(1+\widehat\nu_nT)+\widehat w_n),\ \psi\mapsto \widehat\psi=(\partial+\widehat A_n)\psi,\endalign are defined (Lemma 2.1 and Def. 2.2). Introducing gauge transformations $L\to \overline L$, $\psi\to\overline \psi= g_n^{-1}\psi$, 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 $$(g^k_n-g^k_{n+1})' =e^{g^{k+1}_n-g^k_{n+1}}-e g^k_n -g^{k-1}_{n+1},$$ parametrized by an arbitrary function $g0_0(y)$ and a set of arbitrary constant $r^k$, $k\in\Bbb Z$ can be obtained. Its converse is also shown (Th. 2.7). Parallel results for the two-dimensional discrete hyperbolic Schrödinger operator $$(L\psi)_{n,m} = a_{n,m}\psi_{n,m}+ b_{n,m}\psi_{n+1,m} + c_{n,m}\psi_{n,m+1}+d_{n,m}\psi_{n+1,m+1},$$ are also given (§ 2.2). Then considering the periodic operator of $b_n = -1$, $d_n = 1$, the algebro-geometric spectral theory of {\it B. A. Dubrovin, I. M. Krichever} and {\it 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 [{\it 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 [{\it 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:
 47B39 Difference operators (operator theory) 14H70 Relationships of algebraic curves with integrable systems 35A22 Transform methods (PDE) 35P05 General topics in linear spectral theory of PDE 35Q53 KdV-like (Korteweg-de Vries) equations 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 37K20 Relations of infinite-dimensional systems with algebraic geometry, etc.
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