On numerical study of the discrete spectrum of a two-dimensional Schrodinger operator with soliton potential

TL;DR

This paper numerically investigates the discrete spectrum of 2D Schrödinger operators with soliton potentials, revealing spectral properties and potential blowups related to integrable systems and the Novikov-Veselov equation.

Contribution

It introduces a numerical scheme for analyzing the discrete spectrum of 2D Schrödinger operators with fast decaying potentials, providing evidence for properties of integrable systems.

Findings

Numerical calculation of discrete spectra with soliton potentials.

Spectral properties include multi-dimensional kernels and potential blowups.

Numerical evidence supports theoretical statements on 2D integrable systems.

Abstract

The discrete spectra of certain two-dimensional Schrodinger operators are numerically calculated. These operators have interesting spectral properties, i.e. their kernels are multi-dimensional and the deformations of potentials via the Novikov-Veselov equation (a two-dimensional generalization of the Korteweg-de Vries equation) lead to blowups, and are obtained by the Moutard transformation. The calculations supply the numerical evidence for certain statements on integrable systems related to the 2D Schrodinger operator. The numerical scheme is applicable to a general 2D Schrodinger operator with fast decaying potential.