Laplace transformations and spectral theory of two-dimensionalsemidiscrete and discrete hyperbolic Schrödinger operators

We introduce Laplace transformations of 2D semidiscrete hyperbolic Schrödinger operators and show their relation to a semidiscrete 2D Toda lattice. We develop the algebrogeometric spectral theory of 2D semidiscrete hyperbolic Schrödinger 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 semidiscrete and discrete 2D Toda lattices in terms of Theta-functions.