Consider the transmission eigenvalue problem
$ (\Delta+k^2\mathbf{n}^2) w = 0, \ \ (\Delta+k^2)v = 0\ \ \mbox{in}\ \ \Omega;\quad w = v, \ \ \partial_\nu w = \partial_\nu v\ \ \mbox{on} \ \partial\Omega. $
It is shown in [
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Schematic illustration of
Graphical illustration of the classification of transmission eigenvalues