Abstract
We prove the existence of large regions free of eigenvalues of the interior transmission problem.
Similar content being viewed by others
References
Dimassi M., Sjöstrand J.: Spectral asymptotics in semi-classical limit. London Mathematical Society, Lecture Notes Series, vol. 268. Cambridge University Press, Cambridge (1999)
Dimassi, M., Petkov, V.: Upper bound for the counting function of interior transmission eigenvalues (preprint). arXiv:1308.2594v4 [math.SP]. (2013)
Hitrik M., Krupchyk K., Ola P., Päivärinta L.: The interior transmission problem and bounds of transmission eigenvalues. Math. Res. Lett. 18, 279–293 (2011)
Hörmander L.: The analysis of linear partial differential operators, vol. 3, Pseudo-differential operators. Springer, Berlin (1985)
Lakshtanov E., Vainberg B.: Remarks on interior transmission eigenvalues, Weyl formula and branching billiards. J. Phys. A Math. Theor. 45, 125202 (2012)
Lakshtanov E., Vainberg B.: Bound on positive interior transmission eigenvalues. Inverse Probl. 28, 105005 (2012)
Lakshtanov E., Vainberg B.: Application of elliptic theory to the isotropic interior transmission eigenvalue problem. Inverse Probl. 29, 104003 (2013)
Lakshtanov E., Vainberg B.: Weyl type bound on positive interior transmission eigenvalues. Commun. PDE 39(9), 1729–1740 (2014)
Pham H., Stefanov P.: Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Probl. Imag. 8(3), 795–810 (2014)
Petkov, V. Vodev, G.: Asymptotics of the number of the interior transmission eigenvalues. J. Spectral Theory (to appear)
Robbiano L.: Spectral analysis of interior transmission eigenvalues. Inverse Probl. 29, 104001 (2013)
Robbiano, L.: Counting function for interior transmission eigenvalues (preprint). arXiv:1310.6273 [math.AP] (2013)
Sjöstrand, J.: Singularités analytiques microlocales, Astérisque, vol. 95 (1982)
Sjöstrand J., Vodev G.: Asymptotics of the number of Rayleigh resonances. Math. Ann. 309, 287–306 (1997)
Stefanov P., Vodev G.: Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body. Duke Math. J. 78, 677–714 (1995)
Stefanov P., Vodev G.: Neumann resonances in linear elasticity for an arbitrary body. Commun. Math. Phys. 176, 645–659 (1996)
Sylvester J.: Transmission eigenvalues in one dimension. Inverse Probl. 29, 104009 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Zelditch
Rights and permissions
About this article
Cite this article
Vodev, G. Transmission Eigenvalue-Free Regions. Commun. Math. Phys. 336, 1141–1166 (2015). https://doi.org/10.1007/s00220-015-2311-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-015-2311-2