Bound-state solutions of the Schr\"odinger equation for two novel potentials

TL;DR

This paper presents solutions for bound states in two novel, complex potentials using finite series of Jacobi polynomials, expanding the understanding of non-exactly solvable quantum systems.

Contribution

It introduces two new potential models with rich spectral structures and provides their bound state solutions in terms of Jacobi polynomial series.

Findings

Potential models exhibit complex spectral phase diagrams

Solutions are finite series of Jacobi polynomials

Potential models are not part of the known exactly solvable class

Abstract

We solve the one-dimensional Schr\"odinger equation for the bound states of two potential models with a rich structure as shown by their "spectral phase diagram". These potentials do not belong to the well-known class of exactly solvable problems. The solutions are finite series of square integrable functions written in terms of the Jacobi polynomials.