A Spectral Theory for the Reduced Wave Equation with a Complex Refractive Index

Abstract

over all of R 3, the Euclidean 3-space, where tc is the complex frequency parameter. The n(x) is assumed throughout to be a bounded (measurable) function such that Re 7&(#)^>1 5 Im 71(3;) SjO, Re /&(#)!> Im n(x and 7&(#) — 1 has a compact support. In connection with equation (*) the operator — n~2A will be considered in the Hilbert space H = jL2(R3), square integrable functions over R3, whose norm and inner product will be denoted by || || and ( , ). More explicitly, let HQ be the unique self -adjoint realization of — A in H and N the bounded operator of multiplicatio n by n(x)~2, and consider the unambiguous product H = NHQ with domain identical with that of H0: D(H)=D(Ho), the latter consisting, as is well known, of /GH with second-order L2 derivatives. Together with H it is convenient to consider its adjoint H* = H0N* with D(H*) = N*- 1D(H0). Additive non-self-adjoint perturbations of HQ have been investigated,