A Numerical Treatment of Energy Eigenvalues of Harmonic Oscillators Perturbed by a Rational Function

TL;DR

This paper introduces a numerical method using the double exponential Sinc-collocation approach to accurately compute energy eigenvalues of one-dimensional Schrödinger equations with rational potentials, demonstrating its effectiveness through systematic testing.

Contribution

The paper presents a novel application of DESCM to rational potentials in Schrödinger equations, providing a systematic numerical framework for eigenvalue approximation.

Findings

Accurate eigenvalue approximations for rational potentials.

Effective discretization of Hamiltonian via Sinc expansions.

Validation against known and complex potentials.

Abstract

In the present contribution, we apply the double exponential Sinc-collocation method (DESCM) to the one-dimensional time independent Schr\"odinger equation for a class of rational potentials of the form $V(x) =p(x)/q(x)$. This algorithm is based on the discretization of the Hamiltonian of the Schr\"odinger equation using Sinc expansions. This discretization results in a generalized eigenvalue problem where the eigenvalues correspond to approximations of the energy values of the corresponding Hamiltonian. A systematic numerical study is conducted, beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree.