# Transmission Eigenvalues for Elliptic Operators

@article{Hitrik2010TransmissionEF, title={Transmission Eigenvalues for Elliptic Operators}, author={Michael Hitrik and Katsiaryna Krupchyk and Petri Ola and Lassi P{\"a}iv{\"a}rinta}, journal={SIAM J. Math. Anal.}, year={2010}, volume={43}, pages={2630-2639} }

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-self-adjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.

## 46 Citations

### The interior transmission problem and bounds on transmission eigenvalues

- Mathematics
- 2010

We study the interior transmission eigenvalue problem for sign-definite multiplicative perturbations of the Laplacian in a bounded domain. We show that all but finitely many complex transmission…

### Transmission eigenvalues for degenerate and singular cases

- Mathematics
- 2012

Transmission eigenvalues are the squares of wavenumbers, at which some time harmonic incident wave produces no scattered wave. For the scalar Helmholtz equation, we prove the existence of infinitely…

### On the inverse spectral theory in a non-homogeneous interior transmission problem

- Mathematics
- 2015

We study the inverse spectral problem in an interior transmission eigenvalue problem. The Cartwright’s theory in value distribution theory gives a connection between the distributional structure of…

### The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2018

### Inverse Spectral Problems for Transmission Eigenvalues

- Mathematics
- 2017

We previously encountered transmission eigenvalues and their role in inverse scattering theory in Chap. 6. We now return to this topic and consider the inverse spectral problem for transmission…

### Transmission Eigenvalues

- MathematicsApplied Mathematical Sciences
- 2019

Introduction The study of eigenvalue problems for partial differential equations has a long history during which a variety of themes has emerged. Although historically such efforts have focused on…

### Bounds on positive interior transmission eigenvalues

- Mathematics
- 2012

This paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the…

### Discreteness of Transmission Eigenvalues for Higher-Order Main Terms and Perturbations

- MathematicsSIAM J. Math. Anal.
- 2016

Sylvester's approach via upper triangular compact operators to establish the discreteness of transmission eigenvalues for higher- order main terms and higher-order perturbations is extended.

### Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

- Mathematics
- 2012

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic…

### Transmission Eigenvalues in Inverse Scattering Theory

- Mathematics
- 2012

This survey aims to present the state of the art of research on the transmission eigenvalue problem focussing on three main topics, namely the discreteness of transmission eigenvalues, the existence…

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We study the interior transmission eigenvalue problem for sign-definite multiplicative perturbations of the Laplacian in a bounded domain. We show that all but finitely many complex transmission…

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We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and…

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It is proved the existence of an infinite discrete set of transmission eigenvalues provided that the two contrasts are of opposite signs.

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