The inside–outside duality for scattering problems by inhomogeneous media

Andreas Kirsch, Armin Lechleiter
2013 Inverse Problems  
This paper investigates the relationship between interior transmission eigenvalues k 0 > 0 and the accumulation point 1 of the eigenvalues of the scattering operator S(k) when k approaches k 0 . As it is well known, the spectrum of S(k) is discrete, the eigenvalues µ n (k) lie on the unit circle in C and converge to 1 from one side depending on the sign of the contrast. Under certain (implicit) conditions on the contrast it is shown that interior transmission eigenvalues k 0 can be
more » ... can be characterized by the fact that one eigenvalue of S(k) converges to 1 from the opposite side if k tends to k 0 from below. The proof uses the Cayley transform, Courant's maximum-minimum principle, and the factorization of the far field operator. For constant contrasts that are positive and large enough or negative and small enough, we show that the conditions necessary to prove this characterization are satisfied at least for the smallest transmission eigenvalue.
doi:10.1088/0266-5611/29/10/104011 fatcat:u4rjfwvpxzezppbephwbgyrpve