Exactly solvable two-dimensional Schrodinger operators and Laplace transformations

TL;DR

This paper explores sequences of Laplace transformations applied to 2D Schrödinger operators in periodic fields, leading to exactly solvable models with unique spectral features, including double-periodic operators with algebraic Fermi surfaces.

Contribution

It introduces new exactly solvable 2D Schrödinger operators via Laplace transformations, expanding understanding of their spectral properties and discretizations.

Findings

Identification of exactly solvable 2D Schrödinger operators with nonstandard spectra

Construction of discretizations of 2D operators and transformations

Development of nonstandard 1D discrete operators with unique spectral features

Abstract

Different cases of sequences of the Laplace Transformations for the 2D Schrodinger operator in the periodic magnetic field and electric potential are considered. They lead to the exactly solvable operators with nonstandard spectral properties including the double-periodic operators with algebraic Fermi surface known from the periodic soliton theory. Two appendices are added. In the Appendix I (the author - S. Novikov) two discretizations of the 2D operators and Laplace transformations are constructed. In the Appendix II (the authors - S. Novikov and I. Taimanov) some nonstandard 1D discrete operators are constructed with very interesting spectral properties.