Concentration and non-concentration of eigenfunctions of second-order elliptic operators in divergence form in layered media

Let Ω^′⊂R^d,d=1,2,… be an open bounded smooth domain, and (formula presented) The coordinates in Ω are designated as x=(x^′,y)∈Ω^′×(0,H)[jls-end-space/].The paper deals with the concentration (and non-concentration) properties (in sectors of Ω) of the eigenfunctions of the self-adjoint second-order elliptic operator (formula presented) The coefficient c˜>0 is assumed to be bounded, but no continuity assumption is imposed. It is analogous to the square of the speed of sound in the wave equation, and c˜ is commonly known in the physical literature as the celerity. This study deals with layered media, namely, c˜(x) depends only on the single spatial coordinate y∈(0,H)[jls-end-space/], so that c˜(x)=c˜(x^′,y)=c(y)[jls-end-space/].The eigenvalues of A are partitioned (apart from a small residual set) into two disjoint infinite sets. The corresponding eigenfunctions are labeled as FG (guided) and FNG (non-guided). Their asymptotic properties are expressed by suitable estimates as the associated eigenvalues tend to infinity. The eigenfunctions in FG concentrate in “wells” of c(y)[jls-end-space/], subject to polynomial rate of decay away from the concentration sector. The non-concentrating eigenfunctions in FNG are oscillatory in every sector with non-decaying amplitudes. These results hold uniformly for families of celerities with a common bound on their total variation.The paper leaves as an open problem the question of non-concentration in the case of a function c(y) which is continuous but not of bounded variation.