Advertisement

Transmission Eigenvalues

  • David Colton
  • Rainer Kress
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 93)

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.

References

  1. 3.
    Agmon, S.: Lectures on Elliptic Boundary Value Problems. AMS Chelsea Publishing, Providence 2010.CrossRefzbMATHGoogle Scholar
  2. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  3. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  4. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  5. 31.
    Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. 436, 3839–3863 (2012).CrossRefMathSciNetzbMATHGoogle Scholar
  6. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  7. 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).zbMATHGoogle Scholar
  8. 52.
    Cakoni, F., Colton, D., and Gintides, D.: The interior transmission eigenvalue problem,. SIAM J. Math. Anal. 42, 2912–2921 (2010).CrossRefMathSciNetzbMATHGoogle Scholar
  9. 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).Google Scholar
  10. 55.
    Cakoni, F., Colton, D. and Haddar, H.: The interior transmission problem for regions with cavities. SIAM J. Math. Anal. 42, 145–162 (2010).CrossRefMathSciNetzbMATHGoogle Scholar
  11. 56.
    Cakoni, F., Colton, D., and Haddar, H.: The interior transmission eigenvalue problem for absorbing media. Inverse Problems 28, 045005 (2012).CrossRefMathSciNetzbMATHGoogle Scholar
  12. 57.
    Cakoni, F., Colton, D., and Haddar, H.: Inverse Scattering Theory and Transmission Eigenvalues. SIAM, Philadelphia 2016.CrossRefzbMATHGoogle Scholar
  13. 58.
    Cakoni, F., Colton, D., Meng, S., and Monk, P.: Stekloff eigenvalues in inverse scattering. SIAM J. Appl. Math. 76, 1737–763 (2016).CrossRefMathSciNetzbMATHGoogle Scholar
  14. 59.
    Cakoni, F., Colton, D., and Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering. SIAM Publications, Philadelphia, 2011.CrossRefzbMATHGoogle Scholar
  15. 60.
    Cakoni, F., Colton, D., Monk, P., and Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Problems 26, 07004 (2010).MathSciNetzbMATHGoogle Scholar
  16. 61.
    Cakoni, F., and Gintides, D.: New results on transmission eigenvalues. Inverse Problems and Imaging 4, 39–48 (2010).CrossRefMathSciNetzbMATHGoogle Scholar
  17. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  18. 63.
    Cakoni, F., and Haddar, H.: On the existence of transmission eigenvalues in an inhomogeneous medium. Applicable Analysis 88, 475–493 (2009).CrossRefMathSciNetzbMATHGoogle Scholar
  19. 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).MathSciNetzbMATHGoogle Scholar
  20. 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).MathSciNetzbMATHGoogle Scholar
  21. 67.
    Cakoni, F., and Kirsch, A.: On the interior transmission eigenvalue problem. Int. Jour. Comp. Sci. Math. 3, 142–16 (2010).CrossRefMathSciNetzbMATHGoogle Scholar
  22. 69.
    Cakoni, F. and Kress, R.: A boundary integral equation method for the transmission eigenvalue problem. Applicable Analysis 96, 23–38 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  23. 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.Google Scholar
  24. 73.
    Camaño, J., Lackner, C., and Monk, P.: Electromagnetic Stekloff eigenvalues in inverse scattering. SIAM J. Math. Anal. 49, 4376–4401 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  25. 82.
    Cogar, S: A modified transmission eigenvalue problem for scattering by a partially coated crack. Inverse Problems 34, 115003 (2018).CrossRefMathSciNetzbMATHGoogle Scholar
  26. 83.
    Cogar, S., Colton, D., and Leung, Y.J.: The inverse spectral problem for transmission eigenvalues. Inverse Problems 33, 055015 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  27. 84.
    Cogar, S., Colton, D., Meng, S., and Monk, P.: Modified transmission eigenvalues in inverse scattering theory. Inverse Problems 33, 125002 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  28. 104.
    Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.CrossRefzbMATHGoogle Scholar
  29. 106.
    Colton, D., and Leung, Y.J.: Complex eigenvalues and the inverse spectral problem for transmission eigenvalues. Inverse Problems 29, 104008 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  30. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  31. 108.
    Colton, D., and Leung, Y.J.: The existence of complex transmission eigenvalues for spherically stratified media. Applicable Analysis 96, 39–47 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  32. 109.
    Colton, D., Leung, Y.J., and Meng, S.: Distribution of complex transmission eigenvalues for spherically stratified media. Inverse Problems 31, 035006 (2015).CrossRefMathSciNetzbMATHGoogle Scholar
  33. 124.
    Colton, D., Päivärinta, L., and Sylvester, J.: The interior transmission problem. Inverse Problems and Imaging 1, 13–28 (2007).CrossRefMathSciNetzbMATHGoogle Scholar
  34. 128.
    Cooke, R.: Almost periodic functions. Amer. Math. Monthly 88, 515–526 (1981).CrossRefMathSciNetzbMATHGoogle Scholar
  35. 129.
    Cossonnière, A. and Haddar, H.: Surface integral formulation of the interior transmission problem. J. Integral Equations Applications 25, 341–376 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  36. 139.
    Erdélyi, A.: Asymptotic Expansions. Dover Publications, New York 1956.zbMATHGoogle Scholar
  37. 140.
    Faierman, M. : The interior transmission problem: spectral theory. SIAM J. Math. Anal. 46, 803–819 (2014).CrossRefMathSciNetzbMATHGoogle Scholar
  38. 154.
    Griesmaier, R., and Harrach, B.: Monotonicity in inverse medium scattering on unbounded domains. SIAM J. Appl. Math. 78, 2533–2557 (2018).CrossRefMathSciNetzbMATHGoogle Scholar
  39. 157.
    Guo, Y., Monk, P., and Colton, D.: Toward a time domain approach to the linear sampling method. Inverse Problems 29, 095016 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  40. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  41. 192.
    Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L.: Transmission eigenvalues for elliptic operators. SIAM J. Math. Anal. 43, 2630–2639 (2011).CrossRefMathSciNetzbMATHGoogle Scholar
  42. 219.
    Ji, X., and Sun, J.: A multi-level method for transmission eigenvalues of anisotropic media. J. Comput. Phys. 255, 422–435 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  43. 221.
    Ji, X., Sun, J., and Xie, H.: A multigrid method for Helmholtz transmission eigenvalue problems. J. Sci. Comput., 60, 276–294 (2014).CrossRefMathSciNetzbMATHGoogle Scholar
  44. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  45. 238.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. 2nd ed, Springer, Berlin 2011.CrossRefzbMATHGoogle Scholar
  46. 241.
    Kirsch, A.: On the existence of transmission eigenvalues. Inverse Problems and Imaging 3, 155–172 (2009).CrossRefMathSciNetzbMATHGoogle Scholar
  47. 250.
    Kirsch, A., and Lechleiter, A.: The inside-outside duality for scattering problems by inhomogeneous media. Inverse Problems 29, 104011 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  48. 252.
    Kleefeld, A.: A numerical method to compute interior transmission eigenvalues. Inverse Problems 29, 104012 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  49. 257.
    Koosis, P.: The Logarithmic Integral I. Cambridge University Press, Cambridge 1998.zbMATHGoogle Scholar
  50. 268.
    Kress, R.: Linear Integral Equations. 3rd ed, Springer, Berlin 2014.CrossRefzbMATHGoogle Scholar
  51. 270.
    Kress, R.: Nonlocal impedance conditions in direct and inverse obstacle scattering. Inverse Problems 35, 024002 (2019).CrossRefMathSciNetzbMATHGoogle Scholar
  52. 285.
    Lakshtanov, E., and Vainberg, B.: Bounds on positive interior transmission eigenvalues. Inverse Problems 28, 105005 (2012).CrossRefMathSciNetzbMATHGoogle Scholar
  53. 288.
    Lakshtanov, E., and Vainberg, B.: Weyl type bound on positive interior transmission eigenvalues. Comm. Partial Differential Equations 39, 1729–1740 (2014).CrossRefMathSciNetzbMATHGoogle Scholar
  54. 291.
    Lax, P.D.: Functional Analysis. Wiley-Interscience, New York 2002.zbMATHGoogle Scholar
  55. 294.
    Lechleiter, A. and Peters, S.: The inside-outside duality for inverse scattering problems with near field data. Inverse Problems 31, 085004 (2015).CrossRefMathSciNetzbMATHGoogle Scholar
  56. 302.
    Leung, Y.J., and Colton, D.: Complex transmission eigenvalues for spherically stratified media. Inverse Problems 28, 07505 (2012).CrossRefMathSciNetzbMATHGoogle Scholar
  57. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  58. 345.
    Päivärinta, L., and Sylvester, J.: Transmission eigenvalues. SIAM J. Math. Anal. 40, 738–758 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  59. 347.
    Petrov, V., and Vodev, G.: Asymptotics of the number of the interior transmission eigenvalues. Jour. Spectral Theory 7, 1–31 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  60. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  61. 367.
    Rahman, Q.J., and Schmeisser, G.: Analytic Theory of Polynomials. Clarendon Press, Oxford 2000.zbMATHGoogle Scholar
  62. 375.
    Ringrose, J.R.: Compact Non–Self Adjoint Operators. Van Nostrand Reinhold, London 1971.zbMATHGoogle Scholar
  63. 377.
    Robbiano, L.: Spectral analysis of the interior transmission eigenvalue problem. Inverse Problems 29, 104001 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  64. 378.
    Robbiano, L.: Counting function for interior transmission eigenvalues. Math. Control Related Fields 6, 167–183 (2016).CrossRefMathSciNetzbMATHGoogle Scholar
  65. 381.
    Rundell, W., and Sacks, P.: Reconstruction techniques for classical inverse Sturm–Liouville problems. Math. Comp. 58, 161–183 (1992).CrossRefMathSciNetzbMATHGoogle Scholar
  66. 400.
    Sun, J.: Estimation of transmission eigenvalues and the index of refraction from Cauchy data. Inverse Problems 27, 015009 (2011).CrossRefMathSciNetzbMATHGoogle Scholar
  67. 401.
    Sun, J.: Iterative methods for transmission eigenvalues. SIAM J. Numerical. Anal., 49, 1860–1874 (2011).CrossRefMathSciNetzbMATHGoogle Scholar
  68. 405.
    Taylor, M.E.: Partial Differential Equations. 2nd ed, Springer, New York 2011.zbMATHGoogle Scholar
  69. 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).MathSciNetzbMATHGoogle Scholar
  70. 416.
    Vodev, G.: Parabolic transmission eigenvalue free regions in the degenerate isotropic case. Asymptotic Analysis 108, 147–168 (2018).CrossRefMathSciNetzbMATHGoogle Scholar
  71. 420.
    Wei, G., and Xu, H.: Inverse spectral analysis for the transmission eigenvalue problem. Inverse Problems 29, 115012 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  72. 432.
    Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, San Diego 2001.zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Colton
    • 1
  • Rainer Kress
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August-Universität GöttingenGöttingenGermany

Personalised recommendations