Complex transmission eigenvalues for spherically stratified media (Q2905071)

The transmission eigenvalue problem for a spherically stratified medium in \(\mathbb{R}^d\), \(d=2,3\), is to find a nontrivial solution \(v,w\) of NEWLINE\[NEWLINE\Delta w+k^2 \eta w =0,\;\Delta v+k^2v = 0 \text{ in } B; \;w=v, \partial_\nu w = \partial_\nu v \text{ on } \partial B , \tag{1}NEWLINE\]NEWLINE where \(B = \{| x| < 1\}\) and the index of refraction \(\eta\) only depends on \(| x|\). When \(v\) and \(w\) only depend on \(| x|\), (1) reduces to a coupled set of ordinary differential equations. In this case, the authors prove the following results on the existence of complex transmission eigenvalues. For \(d=2\) and a constant index of refraction, there is an infinite number of complex eigenvalues. For \(d=3\) and constant \(\eta \in \mathbb{N}\), there are no complex eigenvalues, whereas such eigenvalues can exist if \(\eta\) is a rational number. Moreover, complex eigenvalues may exist for a spherically stratified variable index of refraction.