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
, , such theory is given.
For this purpose, introducing the shift operator , the authors rewrite (1) as
Then the first and the second type Laplace transformations
are defined (Lemma 2.1 and Def. 2.2).
Introducing gauge transformations , , 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
parametrized by an arbitrary function and a set of arbitrary constant , can be obtained. Its converse is also shown (Th. 2.7). Parallel results for the two-dimensional discrete hyperbolic Schrödinger operator
are also given (§ 2.2).
Then considering the periodic operator of , , 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.