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

摘要

We introduce Laplace transformations of 2D semi-discrete hyperbolicSchroedinger operators and show their relation to a semi-discrete 2D Todalattice. We develop the algebro-geometric spectral theory of 2D semi-discretehyperbolic Schroedinger operators and solve the direct spectral problem for 2Ddiscrete ones (the inverse problem for discrete operators was already solved byKrichever). Using the spectral theory we investigate spectral properties of theLaplace transformations of these operators. This makes it possible to findsolutions of the semi-discrete and discrete 2D Toda lattices in terms oftheta-functions.