Inverse Sturm-Liouville problems using multiple spectra

Abstract

<p>An eigenvalue problem for a Sturm-Liouville differential operator containing a parameter function and being studied on a given domain is a model for the infinitesimal, vertical vibration of a string of negligible mass, with the ends subject to various constraints. The parameter function of the Sturm-Liouville operator encodes information about the string (its density), and the eigenvalues of the same operator are the squares of the natural frequencies of oscillation of the string. In an inverse Sturm-Liouville problem one has knowledge about the spectral data of the operator and tries to recover the parameter function of the same operator. This thesis deals with the recovery of the parameter function of a Sturm-Liouville operator from knowledge of three sets of eigenvalues. The recovery is achieved theoretically and numerically in two different situations: (a) when the three sets correspond respectively to the vibration of the whole string fixed only at the end points, and the vibrations of each individual piece obtained by fixing the string at an interior node; (b) when the three sets correspond respectively to the vibration of the whole string fixed only at the end points, and the vibrations of each individual piece obtained by attaching the string at an interior node to a spring with a known stiffness constant. Situations when existence or uniqueness of the parameter function is lost are also presented.</p>