General solution of the Schrodinger equation for some trigonometric potentials

Date

2020-05-01

Author

Alıcı, Haydar
Tanriverdi, T.

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In this article, we recursively obtain the general solution of the Schrodinger equation y nu ''(x;lambda)+[lambda-nu(nu+1)v(x)]y nu(x;lambda)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_{\nu }''(x;\lambda )+[\lambda -\nu (\nu +1)v(x)]y_{\nu }(x;\lambda )=0$$\end{document} for non negative integer values of nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu$$\end{document} and an arbitrary values of the eigenvalue parameter lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =n$$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} and non negative integer n. Moreover, the eigenpairs with the homogeneous Drichlet and Neumann boundary conditions are also derived from the general solution.

URI

https://hdl.handle.net/11511/115708

Collections

Department of Mathematics, Article