The inside-outside duality for scattering problems by inhomogeneous media

被引：10

作者：

机构：

[1] Department of Mathematics, Karlsruhe Institute of Technology (KIT)

[2] Center for Industrial Mathematics, University of Bremen

来源：

| 1600年 / IOP Publishing Ltd卷 / 29期

关键词：

Accumulation points - Cayley transforms - Inhomogeneous media - Interior transmission eigenvalues - Minimum Principles - Scattering operators - Scattering problems - Transmission eigenvalue;

D O I：

10.1088/0266-5611/29/10/104011

中图分类号：

学科分类号：

摘要：

This paper investigates the relationship between interior transmission eigenvalues k0 > 0 and the accumulation point 1 of the eigenvalues of the scattering operator when k approaches k0. As is well known, the spectrum of is discrete, the eigenvalues μn(k) lie on the unit circle in and converge to 1 from one side depending on the sign of the contrast. Under certain (implicit) conditions on the contrast it is shown that interior transmission eigenvalues k0 can be characterized by the fact that one eigenvalue of converges to 1 from the opposite side if k tends to k0 from below. The proof uses the Cayley transform, Courant's maximum-minimum principle, and the factorization of the far field operator. For constant contrasts that are positive and large enough or negative and small enough, we show that the conditions necessary to prove this characterization are satisfied at least for the smallest transmission eigenvalue. © 2013 IOP Publishing Ltd.

引用

共 19 条

[11]

Dietz B., Eckmann J.-P., Pillet C.-A., Smilansky U., Ussishkin I., Inside-outside duality for planar billiards: A numerical study, Phys. Rev., pp. 4222-4234, (1995)

[12]

Eckmann J.-P., Pillet C.-A., Spectral duality for planar billiards, Commun. Math. Phys., 170, pp. 283-313, (1995)

[13]

Kirsch A., The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37, pp. 213-225, (1986)

[14]

Kirsch A., Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14, 6, pp. 1489-1512, (1998)

[15]

Kirsch A., On the existence of transmission eigenvalues, Inverse Problems Imaging, 3, pp. 155-172, (2009)

[16]

Kirsch A., Grinberg N.I., The Factorization Method for Inverse Problems, (2008)

[17]

McLaughlin J., Polyakov P., On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differ. Eqns, 107, pp. 351-382, (1994)

[18]

Paivarinta L., Sylvester J., Transmission eigenvalues, SIAM J. Math. Anal., 40, pp. 738-753, (2008)

[19]

Blasten E., Paivarinta L., Sylvester J., Do Corners Always Scatter?, (2012)

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