Exact solvability of two new 3D and 1D nonrelativistic potentials within the TRA framework

TL;DR

This paper introduces two new confined potentials in 1D and 3D, solves them exactly using the TRA with Jacobi polynomial basis, and computes their eigen-energies for specific parameters.

Contribution

It presents the first exact solutions for these new potentials using the TRA and introduces new orthogonal polynomials for the expansion coefficients.

Findings

Exact solutions for the new potentials are obtained.

Eigen-energies are numerically computed for specific parameters.

The approach utilizes Jacobi polynomials and recently introduced orthogonal polynomials.

Abstract

This work is concerned about introducing two new 1D and 3D confined potentials and present their solutions using the Tridiagonal Representation Approach (TRA). The wavefunction is written as a series in terms of square integrable basis functions which are expressed in terms of Jacobi polynomials. Moreover, the expansion coefficients are written in terms of new orthogonal polynomials that were introduced recently by Alhaidari, the analytical properties of these polynomials are yet to be derived. Moreover, we have computed the numerical eigen-energies for both potentials by considering specific choices of the potential parameters.