Twodimensional Periodic Schrödinger Operators Integrable at Energy Eigenlevel
Abstract
The main goal of the first part of the paper is to show that the Fermi curve of a twodimensional periodic Schrödinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Shrödinger 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 NovikovVeselov construction.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 DOI:
 10.48550/arXiv.1903.01778
 arXiv:
 arXiv:1903.01778
 Bibcode:
 2019arXiv190301778I
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 To appear in Functional Analysius and Its Applications, vol. 53, no 1, 2019