General solution of the Schrödinger equation for some trigonometric potentials

Abstract

In this article, we recursively obtain the general solution of the Schrödinger equation $y_{\nu }''(x;\lambda )+[\lambda -\nu (\nu +1)v(x)]y_{\nu }(x;\lambda )=0$ for non negative integer values of $\nu$ and an arbitrary values of the eigenvalue parameter $\lambda$ where v(x) is certain trigonometric potentials. The recursions are obtained from the contour integral solutions of Tanriverdi. By using these contour integral solutions the author have obtained the first few solutions when $\nu =n$, a non negative integer, by means of residue calculations which becomes considerably troublesome or almost impossible for larger values of n. Therefore, the recursive procedure of the present article can be seen as a superior alternative to the method of residue calculation for deriving the general solution for arbitrary values of $\lambda$ and non negative integer n. Moreover, the eigenpairs with the homogeneous Drichlet and Neumann boundary conditions are also derived from the general solution.