The Interior Transmission Eigenvalue Problem

@article{Cakoni2010TheIT,
  title={The Interior Transmission Eigenvalue Problem},
  author={Fioralba Cakoni and David Colton and Drossos Gintides},
  journal={SIAM J. Math. Anal.},
  year={2010},
  volume={42},
  pages={2912-2921}
}
We consider the inverse problem of determining the spherically symmetric index of refraction $n(r)$ from a knowledge of the corresponding transmission eigenvalues (which can be determined from field pattern of the scattered wave). We also show that for constant index of refraction $n(r)=n$, the smallest transmission eigenvalue suffices to determine n, complex eigenvalues exist for n sufficiently small and, for homogeneous media of general shape, determine a region in the complex plane where… 

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References

SHOWING 1-10 OF 16 REFERENCES
New results on transmission eigenvalues
We consider the interior transmission eigenvalue problem corresponding to the inverse scattering problem for an isotropic inhomogeneous medium. We first prove that transmission eigenvalues exist for
On the existence of transmission eigenvalues
The investigation of the far field operator and the Factorization Method in inverse scattering theory leads naturally to the study of corresponding interior transmission eigenvalue problems. In
On the Uniqueness of a Spherically Symmetric Speed of Sound from Transmission Eigenvalues
Abstract Consider the inverse acoustic scattering problem for spherically symmetric inhomogeneity of compact support. Define the corresponding homogeneous and inhomogeneous interior 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
The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data
We consider the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous, anisotropic dielectric medium and show that under certain as- sumptions a lower bound on the norm
Reconstruction of a Spherically Symmetric Speed of Sound
TLDR
In the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented and completely determines the sound of the sound.
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
Integral equation methods in scattering theory
Preface to the Classics Edition Preface Symbols 1. The Riesz-Fredholm theory for compact operators 2. Regularity properties of surface potentials 3. Boundary-value problems for the scalar Helmholtz
Inverse Acoustic and Electromagnetic Scattering Theory
Introduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- III-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle
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