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
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.
AMS-LaTeX, 34 pages; v.2: minor corrections (typos, bibliography etc); v.3: minor corrections
2D semidiscrete hyperbolic Schrödinger operators, 14H70, 39A70, 34K06, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), FOS: Physical sciences, Linear difference operators, General topics in linear spectral theory for PDEs, Mathematical Physics (math-ph), KdV equations (Korteweg-de Vries equations), Transform methods (e.g., integral transforms) applied to PDEs, algebrogeometric spectral theory, theta-functions, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems, semidiscrete 2D Toda lattice, Mathematical Physics
2D semidiscrete hyperbolic Schrödinger operators, 14H70, 39A70, 34K06, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), FOS: Physical sciences, Linear difference operators, General topics in linear spectral theory for PDEs, Mathematical Physics (math-ph), KdV equations (Korteweg-de Vries equations), Transform methods (e.g., integral transforms) applied to PDEs, algebrogeometric spectral theory, theta-functions, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems, semidiscrete 2D Toda lattice, Mathematical Physics
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