Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation. A functional analytic approach

Let Omega(i), Omega(o) be bounded open connected subsets of R-n that contain the origin. Let Omega(epsilon) equivalent to Omega(o) \ c (Omega) over bar (i) for small epsilon > 0. Then we consider a linear transmission problem for the Helmholtz equation in the pair of domains epsilon Omega(i) and Omega(epsilon) with Neumann boundary conditions on partial derivative Omega(o). Under appropriate conditions on the wave numbers in epsilon Omega(i) and Omega(epsilon) and on the parameters involved in the transmission conditions on epsilon partial derivative Omega(i), the transmission problem has a unique solution (u(i)(epsilon, .); u(o) (epsilon, .)) for small values of epsilon > 0. Here u(i)(epsilon, .) and u(o) (epsilon, .) solve the Helmholtz equation in epsilon Omega(i) and Omega(epsilon), respectively. Then we prove that if xi is an element of(Omega(i)) over bar and xi is an element of R-n\Omega(i) then the rescaled solutions u(i) (epsilon, epsilon xi) and u(o) (epsilon, epsilon xi) can be expanded into a convergent power expansion of epsilon, kappa(n) is an element of log epsilon, delta(2,n) log(-1) epsilon, kappa(n) is an element of log(2) epsilon for epsilon small enough. Here kappa(n) = 1 if n is even and kappa(n) = 0 if n is odd and delta(2,2) equivalent to 1 and delta(2,n) equivalent to 0 if n >= 3.