Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

10. Transmission Eigenvalues

verfasst von : David Colton, Rainer Kress

Erschienen in: Inverse Acoustic and Electromagnetic Scattering Theory

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The transmission eigenvalue problem was previously introduced in Sect. 8.​4 where it was shown to play a central role in establishing the completeness of the set of far field patterns in \(L^2(\mathbb {S}^2)\). It was then shown in Sect. 8.​6 that the set of transmission eigenvalues was either empty or formed a discrete set, thus leading to the conclusion that except possibly for a discrete set of values of the wave number k > 0, the set of far field patterns is complete in \(L^2(\mathbb {S}^2)\). In this chapter we return to the subject of transmission eigenvalues and consider further topics of interest. In particular, we begin by showing the existence of transmission eigenvalues and then deriving a monotonicity result for the first positive transmission eigenvalue. We then proceed to describe a boundary integral equation approach to the transmission eigenvalue problem, the existence of complex transmission eigenvalues in the case of a spherically stratified medium, and the inverse spectral problem for the case of such a medium. We conclude this chapter by considering a modified transmission eigenvalue problem in which the wave number k > 0 is kept fixed and the eigenparameter is now an artificial coefficient introduced through the use of a modified far field operator. Our analysis is restricted to the case of acoustic waves.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Electromagnetic Waves in an Inhomogeneous Medium
Nächstes Kapitel The Inverse Medium Problem
Literatur
3.
Agmon, S.: Lectures on Elliptic Boundary Value Problems. AMS Chelsea Publishing, Providence 2010.CrossRefMATH
5.
Aktosun, T., Gintides, D., and Papanicolaou, V.: The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation. Inverse Problems 27, 115004 (2011).CrossRefMathSciNetMATH
6.
Aktosun, T., and Papanicolaou, V.: Reconstruction of the wave speed from transmission eigenvalues for the spherically symmetric variable-speed wave equation. Inverse Problems 29, 065007 (2013).CrossRefMathSciNetMATH
21.
Audibert, L., Cakoni, F., and Haddar, H.: New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data. Inverse Problems 33, 125001 (2017).CrossRefMathSciNetMATH
31.
37.
Bonnet-Ben Dhia, A.S., Carvelho, C., and Chesnel, L.: On the use of perfectly matched layers at corners for scattering problems with sign-changing coefficients. J. Comput. Phys. 322, 224–247 (2016).CrossRefMathSciNetMATH
38.
Bonnet-Ben Dhia, A.S., Chesnel, L. and Haddar, H.: On the use of T-coercivity to study the interior transmission eigenvalue problem. C.R. Math. Acad. Sci. Paris 349, 647–651 (2011).MATH
52.
Cakoni, F., Colton, D., and Gintides, D.: The interior transmission eigenvalue problem,. SIAM J. Math. Anal. 42, 2912–2921 (2010).CrossRefMathSciNetMATH
54.
Cakoni, F., Colton, D., and Haddar, H.: On the determination of Dirichlet and transmission eigenvalues from far field data. C. R. Math. Acad. Sci. Paris, Ser. 1 348, 379–383 (2010).
55.
Cakoni, F., Colton, D. and Haddar, H.: The interior transmission problem for regions with cavities. SIAM J. Math. Anal. 42, 145–162 (2010).CrossRefMathSciNetMATH
56.
Cakoni, F., Colton, D., and Haddar, H.: The interior transmission eigenvalue problem for absorbing media. Inverse Problems 28, 045005 (2012).CrossRefMathSciNetMATH
57.
Cakoni, F., Colton, D., and Haddar, H.: Inverse Scattering Theory and Transmission Eigenvalues. SIAM, Philadelphia 2016.CrossRefMATH
58.
Cakoni, F., Colton, D., Meng, S., and Monk, P.: Stekloff eigenvalues in inverse scattering. SIAM J. Appl. Math. 76, 1737–763 (2016).CrossRefMathSciNetMATH
59.
Cakoni, F., Colton, D., and Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering. SIAM Publications, Philadelphia, 2011.CrossRefMATH
60.
Cakoni, F., Colton, D., Monk, P., and Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Problems 26, 07004 (2010).MathSciNetMATH
61.
62.
Cakoni, F., Gintides, D., and Haddar, H.: The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 42, 237–255 (2010).CrossRefMathSciNetMATH
63.
Cakoni, F., and Haddar, H.: On the existence of transmission eigenvalues in an inhomogeneous medium. Applicable Analysis 88, 475–493 (2009).CrossRefMathSciNetMATH
64.
Cakoni, F., Hsiao, G. C., and Wendland, W. L.: On the boundary integral equations method for a mixed boundary value problem of the biharmonic equation. Complex Var. Theory Appl. 50, 681–696 (2005).MathSciNetMATH
66.
Cakoni, F., Ivanyshyn Yaman, O., Kress, R., and Le Louër, F. : A boundary integral equation for the transmission eigenvalue problem for Maxwell’s equation. Math. Meth. Appl. Math. 41, 1316–1330 (2018).MathSciNetMATH
67.
69.
Cakoni, F. and Kress, R.: A boundary integral equation method for the transmission eigenvalue problem. Applicable Analysis 96, 23–38 (2017).CrossRefMathSciNetMATH
70.
Cakoni, F., Monk, P., and Selgas, V.: Analysis of the linear sampling method for imaging penetrable obstacles in the time domain. Analysis and Partial Differential Equations, to appear.
73.
Camaño, J., Lackner, C., and Monk, P.: Electromagnetic Stekloff eigenvalues in inverse scattering. SIAM J. Math. Anal. 49, 4376–4401 (2017).CrossRefMathSciNetMATH
82.
Cogar, S: A modified transmission eigenvalue problem for scattering by a partially coated crack. Inverse Problems 34, 115003 (2018).CrossRefMathSciNetMATH
83.
Cogar, S., Colton, D., and Leung, Y.J.: The inverse spectral problem for transmission eigenvalues. Inverse Problems 33, 055015 (2017).CrossRefMathSciNetMATH
84.
Cogar, S., Colton, D., Meng, S., and Monk, P.: Modified transmission eigenvalues in inverse scattering theory. Inverse Problems 33, 125002 (2017).CrossRefMathSciNetMATH
104.
Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.CrossRefMATH
106.
Colton, D., and Leung, Y.J.: Complex eigenvalues and the inverse spectral problem for transmission eigenvalues. Inverse Problems 29, 104008 (2013).CrossRefMathSciNetMATH
107.
Colton, D., and Leung, Y.J.: On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems and Imaging 10, 369–377 (2016).CrossRefMathSciNetMATH
108.
Colton, D., and Leung, Y.J.: The existence of complex transmission eigenvalues for spherically stratified media. Applicable Analysis 96, 39–47 (2017).CrossRefMathSciNetMATH
109.
Colton, D., Leung, Y.J., and Meng, S.: Distribution of complex transmission eigenvalues for spherically stratified media. Inverse Problems 31, 035006 (2015).CrossRefMathSciNetMATH
124.
Colton, D., Päivärinta, L., and Sylvester, J.: The interior transmission problem. Inverse Problems and Imaging 1, 13–28 (2007).CrossRefMathSciNetMATH
128.
129.
Cossonnière, A. and Haddar, H.: Surface integral formulation of the interior transmission problem. J. Integral Equations Applications 25, 341–376 (2013).CrossRefMathSciNetMATH
139.
Erdélyi, A.: Asymptotic Expansions. Dover Publications, New York 1956.MATH
140.
154.
Griesmaier, R., and Harrach, B.: Monotonicity in inverse medium scattering on unbounded domains. SIAM J. Appl. Math. 78, 2533–2557 (2018).CrossRefMathSciNetMATH
157.
Guo, Y., Monk, P., and Colton, D.: Toward a time domain approach to the linear sampling method. Inverse Problems 29, 095016 (2013).CrossRefMathSciNetMATH
191.
Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L.: Transmission eigenvalues for operators with constant coefficients. SIAM J. Math. Anal. 42, 2965–2986 (2010).CrossRefMathSciNetMATH
192.
Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L.: Transmission eigenvalues for elliptic operators. SIAM J. Math. Anal. 43, 2630–2639 (2011).CrossRefMathSciNetMATH
219.
Ji, X., and Sun, J.: A multi-level method for transmission eigenvalues of anisotropic media. J. Comput. Phys. 255, 422–435 (2013).CrossRefMathSciNetMATH
221.
Ji, X., Sun, J., and Xie, H.: A multigrid method for Helmholtz transmission eigenvalue problems. J. Sci. Comput., 60, 276–294 (2014).CrossRefMathSciNetMATH
234.
Kirsch, A.: Surface gradients and continuity properties for some integral operators in classical scattering theory. Math. Meth. in the Appl. Sci. 11, 789–804 (1989).CrossRefMathSciNetMATH
238.
Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. 2nd ed, Springer, Berlin 2011.CrossRefMATH
241.
250.
Kirsch, A., and Lechleiter, A.: The inside-outside duality for scattering problems by inhomogeneous media. Inverse Problems 29, 104011 (2013).CrossRefMathSciNetMATH
252.
257.
Koosis, P.: The Logarithmic Integral I. Cambridge University Press, Cambridge 1998.MATH
268.
270.
285.
288.
Lakshtanov, E., and Vainberg, B.: Weyl type bound on positive interior transmission eigenvalues. Comm. Partial Differential Equations 39, 1729–1740 (2014).CrossRefMathSciNetMATH
291.
Lax, P.D.: Functional Analysis. Wiley-Interscience, New York 2002.MATH
294.
Lechleiter, A. and Peters, S.: The inside-outside duality for inverse scattering problems with near field data. Inverse Problems 31, 085004 (2015).CrossRefMathSciNetMATH
302.
Leung, Y.J., and Colton, D.: Complex transmission eigenvalues for spherically stratified media. Inverse Problems 28, 07505 (2012).CrossRefMathSciNetMATH
314.
McLaughlin, J., and Polyakov, P.: On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues. J. Diff. Equations 107, 351–382 (1994).CrossRefMathSciNetMATH
345.
347.
Petrov, V., and Vodev, G.: Asymptotics of the number of the interior transmission eigenvalues. Jour. Spectral Theory 7, 1–31 (2017).CrossRefMathSciNetMATH
348.
Pham, H., and Stefanov, P.: Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems and Imaging 8, 795–810 (2014).CrossRefMathSciNetMATH
367.
Rahman, Q.J., and Schmeisser, G.: Analytic Theory of Polynomials. Clarendon Press, Oxford 2000.MATH
375.
Ringrose, J.R.: Compact Non–Self Adjoint Operators. Van Nostrand Reinhold, London 1971.MATH
377.
378.
381.
Rundell, W., and Sacks, P.: Reconstruction techniques for classical inverse Sturm–Liouville problems. Math. Comp. 58, 161–183 (1992).CrossRefMathSciNetMATH
400.
Sun, J.: Estimation of transmission eigenvalues and the index of refraction from Cauchy data. Inverse Problems 27, 015009 (2011).CrossRefMathSciNetMATH
401.
405.
Taylor, M.E.: Partial Differential Equations. 2nd ed, Springer, New York 2011.MATH
415.
Vodev, G.: High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues. Analysis and Partial Differential Equations 11, 213–236 (2018).MathSciNetMATH
416.
Vodev, G.: Parabolic transmission eigenvalue free regions in the degenerate isotropic case. Asymptotic Analysis 108, 147–168 (2018).CrossRefMathSciNetMATH
420.
432.
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, San Diego 2001.MATH
Metadaten
Titel
Transmission Eigenvalues
verfasst von
David Colton
Rainer Kress
Copyright-Jahr
2019
Buch
Inverse Acoustic and Electromagnetic Scattering Theory

Print ISBN: 978-3-030-30350-1

Electronic ISBN: 978-3-030-30351-8

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-30351-8_10