Spectral Stability of Nonnegative Self-Adjoint Operators

The survey is devoted to spectral stability problems for uniformly elliptic differential operators under the variation of the domain and to the accompanying estimates for the difference of the eigenvalues. Two approaches to the problem are discussed in detail. In the first one it is assumed that the domain is transformed by means of a transformation of a certain class and spectral stability with respect to this transformation is investigated. The second approach is based on the notion of a transition operator and allows direct comparison of the eigenvalues on domains which are close in that or other sense.