Numerical calculation of the discrete spectra of one-dimensional Schrödinger operators with point interactions

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Abstract

© 2018 John Wiley & Sons, Ltd. In this paper, we consider one-dimensional Schrödinger operators Sq on R with a bounded potential q supported on the segment [h0, h1] and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in L2(R) defined by the Schrödinger operator Hq=-d2/dx2+q and matrix point conditions at the ends h0, h1. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator Hq. Moreover, we provide closed-form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self-adjoint and nonself-adjoint problems involving general point interactions described in terms of δ- and δ′-distributions.
Original languageAmerican English
JournalMathematical Methods in the Applied Sciences
DOIs
StatePublished - 1 Jan 2018

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Point Interactions
Discrete Spectrum
Eigenvalues and eigenfunctions
Numerical Calculation
Mathematical operators
Operator
Power series
Adjoint Problem
Singular Potential
Characteristic equation
Eigenfunctions
Closed-form
Numerical Examples

Cite this

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title = "Numerical calculation of the discrete spectra of one-dimensional Schr{\"o}dinger operators with point interactions",
abstract = "{\circledC} 2018 John Wiley & Sons, Ltd. In this paper, we consider one-dimensional Schr{\"o}dinger operators Sq on R with a bounded potential q supported on the segment [h0, h1] and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in L2(R) defined by the Schr{\"o}dinger operator Hq=-d2/dx2+q and matrix point conditions at the ends h0, h1. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator Hq. Moreover, we provide closed-form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self-adjoint and nonself-adjoint problems involving general point interactions described in terms of δ- and δ′-distributions.",
author = "V{\'i}ctor Barrera-Figueroa and Rabinovich, {Vladimir S.}",
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AU - Barrera-Figueroa, Víctor

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N2 - © 2018 John Wiley & Sons, Ltd. In this paper, we consider one-dimensional Schrödinger operators Sq on R with a bounded potential q supported on the segment [h0, h1] and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in L2(R) defined by the Schrödinger operator Hq=-d2/dx2+q and matrix point conditions at the ends h0, h1. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator Hq. Moreover, we provide closed-form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self-adjoint and nonself-adjoint problems involving general point interactions described in terms of δ- and δ′-distributions.

AB - © 2018 John Wiley & Sons, Ltd. In this paper, we consider one-dimensional Schrödinger operators Sq on R with a bounded potential q supported on the segment [h0, h1] and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in L2(R) defined by the Schrödinger operator Hq=-d2/dx2+q and matrix point conditions at the ends h0, h1. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator Hq. Moreover, we provide closed-form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self-adjoint and nonself-adjoint problems involving general point interactions described in terms of δ- and δ′-distributions.

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