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Spectral theory of two-dimensional periodic operators and its applications. (English. Russian original) Zbl 0699.35188
Russ. Math. Surv. 44, No. 2, 145-225 (1989); translation from Usp. Mat. Nauk 44, No. 2(266), 121-184 (1989).
A constructive approach to the study of Riemann surfaces of Bloch-type functions is presented. Abstract Bloch-solutions are given using series similar to perturbation ones and their convergence is proved. The spectral theory of finite-zone operators is developed. The completeness of the Bloch series is established. Particular applications are discussed in detail. Chapter 1: Spectral theory of the time dependent Schrödinger operator. Chapter 2: The periodic boundary value problem for the Kadomtsev-Petviashvili equations. Chapter 3: Spectral theory for one- level energy 2-D periodic Schrödinger operator.
Reviewer: N.Vulchanov

35P05 General topics in linear spectral theory for PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35G05 Linear higher-order PDEs
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