Two-dimensional von Neumann--Wigner potentials with a multiple positive eigenvalue

TL;DR

This paper constructs specific two-dimensional Schrödinger operators with smooth, decaying potentials that have multiple positive eigenvalues, using the Moutard transformation method.

Contribution

It introduces a novel application of the Moutard transformation to generate rational and trigonometric potentials with multiple positive eigenvalues in two dimensions.

Findings

Constructed explicit examples of potentials with multiple positive eigenvalues.

Demonstrated the potentials are smooth and decay at infinity.

Potential functions are rational and involve sines and cosines.

Abstract

By the Moutard transformation method we construct two-dimensional Schrodinger operators with real smooth potential decaying at infinity and with a multiple positive eigenvalue. These potentials are rational functions of spatial variables and their sines and cosines.