Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials

✍ Scribed by M. Kobayashi

• Publisher: Elsevier Science
• Year: 1989
• Tongue: English
• Weight: 317 KB
• Volume: 18
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/0898-1221(89)90220-4

✦ Synopsis

Alam'aet--In this paper we consider three examples of discontinuous Sturm-Liouville problems with symmetric potentials. The ¢igcnvalues of the systems were determined using the classical fourth order Runge--Kutta method. These eigenvalues are used to reconstruct the potential function using an algorithm presented in Kobayashi [1,2]. The results of our numerical experiments are discussed.

1. A MATTHIEU EQUATION

In this section we generate the first fifteen eigenvalucs of two discontinuous Sturm-Liouville systems with symmetric boundary and jump conditions, then we try to reconstruct the potential function of the second system, a Matthieu potential, using the fifteen eigenvalues and algorithm from Kobayashi [1,2]. Begin with the Sturm-Liouville system with potential q --0: System I with boundary conditions: and symmetric jump conditions: u(d~ +) = au(d, -), u(d:-) ffi au(d: + ), --U # ~ ~.Z/~ u'(O) ffi u '(n) = 0 u'(d~ +) = a-~u'(d~ -) + bu(d~ -), u'(d:-) ffi a-~u'(d, + ) -bu(d,-),