JournalsjstVol. 7, No. 1pp. 1–31

Asymptotics of the number of the interior transmission eigenvalues

  • Vesselin Petkov

    Université de Bordeaux, Talence, France
  • Georgi Vodev

    Université de Nantes, France
Asymptotics of the number of the interior transmission eigenvalues cover
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Abstract

We prove Weyl asymptotics N(r)=crd+Oϵ(rdκ+ϵ)N(r) = c r^d + {\mathcal O}_{\epsilon}(r^{d - \kappa + \epsilon}), for all 0<ϵ10< \epsilon \ll 1, for the counting function N(r)={λjC{0} ⁣:λjr2}N(r) = \sharp\{\lambda_j \in \mathbb C \setminus \{0\}\colon |\lambda_j| \leq r^2\}, r>1r>1, of the interior transmission eigenvalues (ITE), λj\lambda_j. Here d2d \geq 2 denotes the space dimension and 0<κ10<\kappa\le 1 is such that there are no (ITE) in the region {λC ⁣:ImλC(Reλ+1)1κ2}\{\lambda\in \mathbb C\colon |{\rm Im}\,\lambda|\ge C(|{\mathrm {Re}}\,\lambda|+1)^{1-\frac{\kappa}{2}}\} for some C>0C>0.

Cite this article

Vesselin Petkov, Georgi Vodev, Asymptotics of the number of the interior transmission eigenvalues. J. Spectr. Theory 7 (2017), no. 1, pp. 1–31

DOI 10.4171/JST/154