A Numerical Treatment of Energy Eigenvalues of Harmonic Oscillators Perturbed by a Rational Function
This work provides a numerical method for solving eigenvalue problems in quantum mechanics for a specific class of potentials, but it is an incremental application of an existing technique.
The authors apply the double exponential Sinc-collocation method to compute energy eigenvalues for one-dimensional Schrödinger equations with rational potentials, demonstrating accuracy on test potentials and extending to higher-degree rational potentials.
In the present contribution, we apply the double exponential Sinc-collocation method (DESCM) to the one-dimensional time independent Schrödinger 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 Schrödinger 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.