On the solutions of the Schrödinger equation with 2nd Pöschl–Teller potentials

Abstract

We delve into the intricacies of the hyperbolic Pöschl–Teller potential, often referred to as the 2nd Pöschl–Teller potential (PTP), and explore its implications on the Schrödinger equation for arbitrary l state case. This exploration encompasses a comprehensive review of various methodologies employed in studying the exact solutions of the S wave scenario, along with approximations concerning the centrifugal term within the radial equation for the arbitrary l setup. Our investigation yields exact solutions for the quasi-exact arbitrary l state model. We achieve this by employing a novel approach rooted in Fuchsian differential equations, thereby presenting a potential solution for the S wave case as well. This innovative method holds promise, particularly in seeking solutions for the Schrödinger equation involving non-trivial potentials where exact solutions remain elusive.

Keywords

• Second Pöschl–Teller potential
• Quasi-exact solutions
• Heun differential equation
• Quantization rule