A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications

2018/12/26 by Khmelnytskaya, Kira V., Kravchenko, Vladislav V., Torba, Sergii M. · 1 citation
#34A25 #34A55 #34B24 #34L10 #34L20 #34L40 #35L20 #65L99 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1812.10513

Abstract

The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schrödinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding Sturm-Liouville problem. Since both series converge slowly and in general only in a certain distributional sense we find a way to improve these expansions and make them convergent uniformly and absolutely by adding and subtracting corresponding terms. A numerical illustration of the obtained results is given.

Cited by

• Reconstruction techniques for inverse Sturm-Liouville problems with complex coefficients

Related

• Transmutation operators for Schrödinger equations with distributional potentials and the associated impedance equation
• Standard transmutation operators for the one dimensional Schr "odinger\n operator with a locally integrable potential
• On construction of transmutation operators for perturbed Bessel\n equations
• Calculation of the eigenfunctions and eigenvalues of Schrödinger type equations by asymptotic Taylor expansion method (ATEM)
• Representation of solutions of the one-dimensional Dirac equation in terms of Neumann series of Bessel functions