Journal of Spectral Theory


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Volume 7, Issue 1, 2017, pp. 1–31
DOI: 10.4171/JST/154

Published online: 2017-03-22

Asymptotics of the number of the interior transmission eigenvalues

Vesselin Petkov[1] and Georgi Vodev[2]

(1) Université de Bordeaux, Talence, France
(2) Université de Nantes, France

We prove Weyl asymptotics $N(r) = c r^d + {\mathcal O}_{\epsilon}(r^{d - \kappa + \epsilon})$, for all $0< \epsilon \ll 1$, for the counting function $N(r) = \sharp\{\lambda_j \in \mathbb C \setminus \{0\}\colon |\lambda_j| \leq r^2\}$, $r>1$, of the interior transmission eigenvalues (ITE), $\lambda_j$. Here $d \geq 2$ denotes the space dimension and $0<\kappa\le 1$ is such that there are no (ITE) in the region $\{\lambda\in \mathbb C\colon |{\rm Im}\,\lambda|\ge C(|{\mathrm {Re}}\,\lambda|+1)^{1-\frac{\kappa}{2}}\}$ for some $C>0$.

Keywords: Interior transmission eigenvalues, Weyl formula with remainder, eigenvalue-free regions

Petkov Vesselin, Vodev Georgi: Asymptotics of the number of the interior transmission eigenvalues. J. Spectr. Theory 7 (2017), 1-31. doi: 10.4171/JST/154