Laplace transformations and spectral theory of two-dimensional semi-discrete and discrete hyperbolic Schroedinger operators

TL;DR

This paper develops a spectral theory for 2D semi-discrete and discrete hyperbolic Schrödinger operators, introduces Laplace transformations, and connects these to Toda lattices, enabling solutions via theta-functions.

Contribution

It introduces Laplace transformations for 2D hyperbolic Schrödinger operators and develops their spectral theory, linking to Toda lattices and solving the direct spectral problem for discrete cases.

Findings

Spectral properties of Laplace transformations are characterized.

Solutions to 2D Toda lattices are expressed in theta-functions.

Spectral theory for semi-discrete operators is established.

Abstract

We introduce Laplace transformations of 2D semi-discrete hyperbolic Schroedinger operators and show their relation to a semi-discrete 2D Toda lattice. We develop the algebro-geometric spectral theory of 2D semi-discrete hyperbolic Schroedinger operators and solve the direct spectral problem for 2D discrete ones (the inverse problem for discrete operators was already solved by Krichever). Using the spectral theory we investigate spectral properties of the Laplace transformations of these operators. This makes it possible to find solutions of the semi-discrete and discrete 2D Toda lattices in terms of theta-functions.