Rayleigh–Schrödinger Perturbation of Semigroups

Abstract

A second order differential operator with abstract operator coefficients is considered in the spaces $L^p (0,1),1 \leq p < \infty $, where the operator coefficient and the boundary conditions are analytic functions of a finite number of complex parameters. Asymptotic expressions for the eigenvalues and eigenfunctions are obtained, which are uniform over all eigenvalues. These are applied to develop a structure theory for the operators and the semigroups they generate. An application to the problem of determining the extent to which the semigroup approximates the identity operator is given.