Eigenparameter Dependent Inverse Sturm-Liouville Problems

Abstract

Uniqueness of and numerical techniques for the inverse Sturm-Liouville problem with eigenparameter dependent boundary conditions will be discussed. We will use a Gel'fand-Levitan technique to show that the potential q in

can be uniquely determined using spectral data. In the presence of finite spectral data, q can be reconstructed using a successive approximation method that involves solving a hyperbolic boundary value problem that arises in the the Gel'fand-Levitan analysis. We also consider a shooting method where the right endpoint boundary condition is used in conjunction with a quasi-Newton scheme to recover the unknown potential, q.

Keywords:

• Inverse spectral theory
• Eigenparameter dependent Sturm-Liouville problem
• Gel'fand-Levitan method
• Quasi-Newton method

Acknowledgment

The first author gratefully acknowledges partial support from the Association for Women in Mathematics, the Kentucky National Science Foundation Experimental Program for the Stimulation of Competitive Research (KY NSF EPSCoR) and the National Science Foundation. The second author gratefully acknowledges partial support from the National Science Foundation.