On the determination of Dirichlet or transmission eigenvalues from far field data

Fioralba Cakoni; David Colton; Houssem Haddar

doi:10.1016/j.crma.2010.02.003

On the determination of Dirichlet or transmission eigenvalues from far field data

Fioralba Cakoni, David Colton, Houssem Haddar

Research output: Contribution to journal › Article › peer-review

111 Scopus citations

Abstract

We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for -Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.

Original language	English (US)
Pages (from-to)	379-383
Number of pages	5
Journal	Comptes Rendus Mathematique
Volume	348
Issue number	7-8
DOIs	https://doi.org/10.1016/j.crma.2010.02.003
State	Published - Apr 2010
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics(all)

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10.1016/j.crma.2010.02.003

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title = "On the determination of Dirichlet or transmission eigenvalues from far field data",

abstract = "We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for -Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.",

author = "Fioralba Cakoni and David Colton and Houssem Haddar",

note = "Funding Information: ✩ The research of F.C. and D.C. was supported in part by the U.S. Air Force Office of Scientific Research under Grant FA-9550-08-1-0138. This research was in part supported by the associate team ISIP of INRIA-UDEL. E-mail addresses: cakoni@math.udel.edu (F. Cakoni), colton@math.udel.edu (D. Colton), haddar@cmapx.polytechnique.fr (H. Haddar).",

year = "2010",

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AU - Cakoni, Fioralba

AU - Colton, David

AU - Haddar, Houssem

N1 - Funding Information: ✩ The research of F.C. and D.C. was supported in part by the U.S. Air Force Office of Scientific Research under Grant FA-9550-08-1-0138. This research was in part supported by the associate team ISIP of INRIA-UDEL. E-mail addresses: cakoni@math.udel.edu (F. Cakoni), colton@math.udel.edu (D. Colton), haddar@cmapx.polytechnique.fr (H. Haddar).

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N2 - We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for -Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.

AB - We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for -Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.

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JF - Comptes Rendus Mathematique

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ER -

On the determination of Dirichlet or transmission eigenvalues from far field data

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