On Spectral Properties of the Acoustic Propagator in a Layered Band

Abstract

The acoustic propagator is the self-adjoint operatorH=−∇·c(x)2∇, defined inL2(Q), whereQ⊆R2is a band of finite width, given byQ={(x, z), x∈R, 0⩽z⩽Γ}. The wave velocitycdepends only on the horizontal coordinatex, and is a measurable bounded function, converging toc±>0 asx→±∞. It is proved that the resolvent operatorR(z)=(H−z)−1, Imz≠0, can be extended continuously to the closed upper (or lower) half-plane, in a suitable weighted-L2topology (“Limiting Absorption Principle”). In particular, this continuity holds at the thresholdλ=0. It follows as a corollary thatHhas no point spectrum.