Notes on Transmission Eigenvalues

Here k = ω/co is the wave number, ω denotes the frequency of excitation, S2 = {x ∈ R3, |x| = 1} is the unit sphere in R3, and co is such that its real and imaginary parts are respectively <(co)> 0 and =(co)60. The support ofD is assumed to be such that R3 \D is connected, and that ∂D is of Lipschitz type. As a canonical example of the scattering by a penetrable obstacle, consider next the case where D is characterized by a spatially-varying sound speed c(x) and associated index of refraction, n(x) = (co/c), such that i) <(c)> cD > 0 and =(c)6 0 where cD is a constant, ii) n∈L∞(D), and iii) ∇n is sufficiently small so that it can be omitted from the field equation. For simplicity of exposition, an additional hypothesis is made that the mass density of the system, ρ, is constant throughout (this restriction can however be relaxed, see Remark 2). With such premises the direct