Well-posed Inverse Eigenvalue Problems

Abstract

Summary Inverse eigenvalue problems associated with self-adjoint differential equations of order two or higher are considered. The question of what kind of data are necessary and sufficient to insure the existence of a unique solution is examined. A method of solution for well-posed inverse eigenvalue problems is then presented. The implications of this analysis to the inverse problem for the Earth's spheroidal modes of vibration is also discussed. In this talk, I propose to present a partial survey of the mathematical literature on inverse eigenvalue problems, with an eye toward geophysical applications, and to state some new results which I obtained recently. Perhaps the inverse eigenvalue problem which is best known to geophysicists is that related to the normal modes of vibration of the Earth. It can be stated thus: assuming that the Earth is spherically symmetric and given the natural frequencies ,S, and ,T, of the spheroidal and torsional modes of vibration, can we deduce the density p(r) and the speed of propagation Vr Takeuchi & Saito 1972) and very little is known about inverse eigenvalue problems of order higher than two. In order to gain an understanding of inverse eigenvalue problems one should perhaps divorce oneself from a specific problem and investigate rather a sequence of problems of increasing complexity. If you allow me to adopt this attitude, then the first task would be a rough classification of eigenvalue problems. Without pretending to be exhaustive, let me divide eigenvalue problems into two classes depending upon whether they are associated with ordinary or partial differential equations, i.e. depending upon whether they are related to one-dimensional or multi-dimensional problems. The following Sturm-Liouville problem: u+{n-q(x)}u = 0