Weakly Perturbed Forms and the Spectral Properties of Associated Eigenvalue Problems

Abstract

A bounded sesquilinear form, $a( \cdot , \cdot )$, defined on a Hilbert space X is called a weak perturbation of a Hermitian form, $h( \cdot , \cdot )$, if an operator associated with $a( \cdot , \cdot )$ is a weak perturbation, in the sense of Kren, of a selfadjoint operator associated with $h( \cdot , \cdot )$. Under suitable hypotheses on the forms $a( \cdot , \cdot )$ and $h( \cdot , \cdot )$, it is shown that $a( \cdot , \cdot )$ is a weak perturbation of $h( \cdot , \cdot )$. In this case, results of Keldyš and Kren on the spectral properties of weakly perturbed selfadjoint operators are applied to the eigenvalue problem, $a(u,v) = \lambda (u,v)$, where $( \cdot , \cdot )$ is the inner product on X. In particular, the completeness in X of the space of generalized eigenvectors and various results on the localization and distribution of eigenvalues are demonstrated. Applications of these results to three different kinds of elliptic differential eigenvalue problems are included. Among these applications is a problem from the theory of hydrodynamic stability and a problem in which the eigenvalue parameter appears in a boundary condition.