On the distribution of Born transmission eigenvalues in the complex plane

We analyze an approximate interior transmission eigenvalue problem in ${\mathbb{R}}^d$ for $d=2$ or $d=3$, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. We show the discreteness of transmission eigenvalues in the complex plane. Moreover, using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.