On Spectral Properties of the Acoustic Propagator in a Layered Band

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.