## 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
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### References:

 [1] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “The Schrödinger equation in a magnetic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR, 229 (1976), 15-18; English transl.: Soviet Math. Dokl., 17 (1977), 947-951. · Zbl 0441.35021 [2] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Non-linear Equations of Korteweg-de Vries Type, Finite-Zone Linear Operators, and Abelian Varieties,” Uspekhi Mat. Nauk, 31:1(187) (1976), 55-136; English transi.: Russian Math. Surveys, 31:1 (1976), 59-146. · Zbl 0326.35011 [3] I. M. Krichever, “Potentials with zero coefficient of reflection on a background of finite-zone potentials,” Punkts. Anal. Prilozhen., 9:2 (1975), 77-78; English transi.: Functional Anal. Appl., 9:2 (1975), 161-163. · Zbl 0333.34022 [4] I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funkts. Anal. Prilozhen., 11:1 (1977), 15-31; English transi.: Functional Anal. Appl., 11:1 (1977), 12-26. · Zbl 0368.35022 [5] I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications,” Uspekhi Mat. Nauk, 44:2(266) (1989), 121-184; English transi.: Russian Math. Surveys, 44:2 (1989), 145-225. · Zbl 0699.35188 [6] I. M. Krichever, “Spectral theory of finite-zone nonstationary Schrödinger operators. A non-stationary Peierls model,” Funkts. Anal. Prilozhen., 20:3 (1986), 42-54; English transi.: Functional Anal. Appl., 20:3 (1986), 203-214. [7] S. M. Natanzon, “Nonsingular finite-zone two-dimensional Schrödinger operators and prymians of real curves,” Funkts. Anal. Prilozhen., 22:1 (1988), 79-80; English transi.: Functional Anal. Appl., 22:1 (1988), 68-70. · Zbl 0656.35025 [8] A. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations,” Dokl. Akad. Nauk SSSR, 279:1 (1984), 20-24. · Zbl 0613.35020 [9] A. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional Schrödinger operators. Potential operators,” Dokl. Akad. Nauk SSSR, 279:1 (1984), 784-788; English transi.: Soviet Math. Dokl., 30 (1984), 588-591. · Zbl 0602.35024 [10] I. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces,” Uspekhi Mat. Nauk, 61:1(367) (2006), 85-164; English transi.: Russian Math. Surveys, 61:1 (2006), 79-159.
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