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

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

2003-11-20

arxiv.org/abs/math-ph/0311036v3

Int. Math. Res. Not. 2005, no. 18, 1089--1126

Physics

Mathematical Physics

AMS-LaTeX, 34 pages; v.2: minor corrections (typos, bibliography etc); v.3: minor corrections

Scientific paper

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.

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