Variational Principles for Eigenvalues

Abstract

In this chapter we will study inequalities that are used for localising the spectrum of a Hermitian operator. Such results are motivated by several interrelated considerations. It is not always easy to calculate the eigenvalues of an operator. However, in many scientific problems it is enough to know that the eigenvalues lie in some specified intervals. Such information is provided by the inequalities derived here. While the functional dependence of the eigenvalues on an operator is quite complicated, several interesting relationships between the eigenvalues of two operators A, B and those of their sum A + B are known. These relations are consequences of variational principles. When the operator B is small in comparison to A, then A + B is considered as a perturbation of A or an approximation to A. The inequalities of this chapter then lead to perturbation bounds or error bounds.