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

• Published in: Results in physics Vol. 58; p. 107455
• Main Authors: Martinez-Espinosa, J.M., Balderas-Navarro, R.E., Dong, Shi-Hai
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2024; Elsevier
• Subjects: Heun differential equation; Quantization rule; Quasi-exact solutions; Second Pöschl–Teller potential; Quasi-exact solutions; Heun differential equation; Second Pöschl–Teller potential; Quantization rule
• ISSN: 2211-3797; 2211-3797
• DOI: 10.1016/j.rinp.2024.107455

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. •The quasi exact solutions of the second Pöschl–Teller potential are presented.•Suitable variable transformation leads to the quantization condition.•Confluent Heun differential equation approach fails to ind a quantization condition.•Suitable approximation to centrifugal term is chosen.