Explicit general solution of the squared secant potential and some consequences

Alici H.

RAMANUJAN JOURNAL, vol.62, no.1, pp.111-140, 2023 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 62 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.1007/s11139-023-00748-8
  • Journal Name: RAMANUJAN JOURNAL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.111-140
  • Middle East Technical University Affiliated: No

Abstract

In this article, we obtain an explicit general solution of the Schrodinger equation with the squared secant potential v(v + 1)sec(2)x in terms of elementary functions for non negative integer value of v = n. Alternatively, we provide a general solution in terms of the Gauss hypergeometric function for any parameter value v. Then, we derive some hypergeometric identities by comparing these two sets of solutions when v is a non negative integer. With the help of these hypergeometric formulas, we derive several existing explicit representations as well as new ones for some special functions and orthogonal polynomials including the Legendre, Ferrers, and the Bessel functions, the Jacobi and the Lommel polynomials containing specific parameters.