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

A. N. Adilkhanov; I. A. Taimanov

doi:10.1016/j.cnsns.2016.04.033

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

A. N. Adilkhanov, I. A. Taimanov

National Laboratory Astana

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

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.

Original language	English
Pages (from-to)	83-92
Number of pages	10
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	42
DOIs	https://doi.org/10.1016/j.cnsns.2016.04.033
Publication status	Published - Jan 1 2017

Keywords

Discrete spectrum
Galerkin method
Schrodinger operator
Soliton

ASJC Scopus subject areas

Numerical Analysis
Modelling and Simulation
Applied Mathematics

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title = "On numerical study of the discrete spectrum of a two-dimensional Schr{\"o}dinger operator with soliton potential",

abstract = "The discrete spectra of certain two-dimensional Schr{\"o}dinger 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 Schr{\"o}dinger operator. The numerical scheme is applicable to a general 2D Schr{\"o}dinger operator with fast decaying potential.",

keywords = "Discrete spectrum, Galerkin method, Schrodinger operator, Soliton",

author = "Adilkhanov, {A. N.} and Taimanov, {I. A.}",

note = "Funding Information: This work was supported by the Science Committee of the Ministry of Education and Science of Republic of Kazakhstan (program 0217/ptf-14-OT), RFBR (grant 15-01-01671) and the Government of the Russian Federation (contract 14.Y26.31.0006). ",

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AU - Adilkhanov, A. N.

AU - Taimanov, I. A.

N1 - Funding Information: This work was supported by the Science Committee of the Ministry of Education and Science of Republic of Kazakhstan (program 0217/ptf-14-OT), RFBR (grant 15-01-01671) and the Government of the Russian Federation (contract 14.Y26.31.0006).

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Y1 - 2017/1/1

N2 - 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.

AB - 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.

KW - Discrete spectrum

KW - Galerkin method

KW - Schrodinger operator

KW - Soliton

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