*(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

${b}_{n}\ne 0$, ${d}_{n}\ne 0$, such theory is given.

For this purpose, introducing the shift operator $T{\psi}_{n}\left(y\right)=\psi \left(y\right){\psi}_{n+1}$, 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 $L\to \overline{L}$, $\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

parametrized by an arbitrary function $g{0}_{0}\left(y\right)$ and a set of arbitrary constant ${r}^{k}$, $k\in \mathbb{Z}$ 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 ${b}_{n}=-1$, ${d}_{n}=1$, 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:

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. |