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Il’ina, A. V.; Krichever, I. M.; Nekrasov, N. A.

Two-dimensional periodic Schrödinger operators integrable at an energy eigenlevel. (English. Russian original) Zbl 1427.35161

Funct. Anal. Appl. 53, No. 1, 23-36 (2019); translation from Funkts. Anal. Prilozh. 53, No. 1, 31-48 (2019).
Summary: The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrödinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrödinger equation at the zero energy level is a smooth \(M\)-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrödinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov-Veselov construction.

MSC:

35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
35G20 Nonlinear higher-order PDEs

Keywords:

spectral theory of periodic differential operators; complex Fermi curve; Baker-Akhiezer function; \(M\)-curve
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References:

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