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

Abstract

Abstract The discrete spectra of certain two-dimensional Schrödinger operators are numerically calculated. These operators are obtained by the Moutard transformation and have interesting spectral properties: 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. The calculations supply the numerical evidence for some statements about the integrable systems related to a 2D Schrödinger operator. The numerical scheme is applicable to a general 2D Schrödinger operator with fast decaying potential.