Elliptic finite-band potentials of a non-self-adjoint Dirac operator

TL;DR

This paper introduces a new family of elliptic potentials for a non-self-adjoint Dirac operator, providing a complete spectral characterization and linking it to integrable systems and special functions.

Contribution

It explicitly constructs a two-parameter family of finite-band elliptic potentials and relates spectral problems to Heun's equation and non-self-adjoint tridiagonal operators.

Findings

Spectrum characterized via eigenvalue problems and Heun's equation.

All related tridiagonal operators have real eigenvalues.

Potential solutions generate finite-genus solutions for nonlinear Schrödinger hierarchy.

Abstract

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials for a non-self-adjoint Dirac operator which connects two previously known limiting cases in which the elliptic parameter is zero or one. A full characterization of the spectrum is obtained by relating the periodic and antiperiodic eigenvalue problems for the Dirac operator to corresponding eigenvalue problems for tridiagonal operators acting on Fourier coefficients in a weighted Hilbert space and to appropriate connection problems for Heun's equation. In turn, these problems are related to four non-self-adjoint unbounded tridiagonal operators, all of which nonetheless have only real eigenvalues. For certain parameter values, the corresponding elliptic potentials generate finite-genus solutions for all the positive and negative flows of the focusing nonlinear Schr\"odinger hierarchy.