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. 
View PDF on arXiv
46 Citations
Highly Influential Citations
4
Background Citations
14
Methods Citations
3

46 Citations

The interior transmission problem and bounds on transmission eigenvalues

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

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

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

Inverse Spectral Problems for Transmission Eigenvalues

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

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

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

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

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

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…

References

SHOWING 1-10 OF 24 REFERENCES

Transmission Eigenvalues for Operators with Constant Coefficients

The interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients are studied and a connection to scattering theory is presented.

Non linear eigenvalue problems

In this paper we consider generalized eigenvalue problems for a family of operators with a polynomial dependence on a complex parameter. This problem is equivalent to a genuine non self-adjoint…

The interior transmission problem and bounds on transmission eigenvalues

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…

The Existence of an Infinite Discrete Set of Transmission Eigenvalues

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…

On the interior transmission eigenvalue problem

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

Nonlinear eigenvalues and analytic hypoellipticity

An integral equation approach and the interior transmission problem for Maxwell's equations

In the first part of this paper we recall the direct scattering problem for time harmonic electromagnetic fields where arbitrary incident fields are scattered by a medium described by a space…

Semiclassical analysis of a nonlinear eigenvalue problem and nonanalytic hypoellipticity

A semiclassical analysis of a nonlinear eigenvalue problem arising from the study of the failure of analytic hypoellipticity is given. A general family of hypoelliptic, but not analytic hypoelliptic…

The interior transmission problem

The interior transmission problem is a boundary value problem that plays a basic role in inverse scattering theory but unfortunately does not seem to be included in any existing theory in partial…

Trace ideals and their applications

Preliminaries Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for $\mathcal J_P$ Trace, determinant, and Lidskii's theorem $f(x)g(-i\nabla)$ Fredholm theory…