Upper bound for the counting function of interior transmission eigenvalues

For the complex interior transmission eigenvalues (ITE) we study for small $\theta > 0$ the counting function $$N(\theta, r) = #\{\lambda \in \C:\: \lambda \: {\rm is} \: {\rm (ITE)},\: |\lambda| \leq r, \: 0 \leq \arg \lambda \leq \theta\}.$$ We obtain for fixed $\theta > 0$ an upper bound $N(\theta, r) \leq C r^{n/2}, \: r \geq r(\theta).$

[1]  M. Vishik,et al.  ELLIPTIC PROBLEMS WITH A PARAMETER AND PARABOLIC PROBLEMS OF GENERAL TYPE , 1964 .

[2]  J. Sjöstrand.  Geometric bounds on the density of resonances for semiclassical problems , 1990 .

[3]  A. G. Kostyuchenko,et al.  Spectral asymptotics of polynomial pencils of differential operators in bounded domains , 1991 .

[4]  A. G. Kostyuchenko,et al.  SPECTRAL ASYMPTOTICS OF NONSELFADJOINT ELLIPTIC SYSTEMS OF DIFFERENTIAL OPERATORS IN BOUNDED DOMAINS , 1992 .

[5]  M. Zworski,et al.  Distribution of scattering poles near the real axis , 1992 .

[6]  M. Zworski,et al.  Estimates on the number of scattering poles near the real axis for strictly convex obstacles , 1993 .

[7]  M. Dimassi,et al.  Spectral Asymptotics in the Semi-Classical Limit: Improvement when the periodic trajectories form a set of measure 0 , 1999 .

[8]  Petri Ola,et al.  Transmission Eigenvalues for Operators with Constant Coefficients , 2010, SIAM J. Math. Anal..

[9]  Petri Ola,et al.  The interior transmission problem and bounds on transmission eigenvalues , 2010, 1009.5640.

[10]  Fioralba Cakoni,et al.  The Existence of an Infinite Discrete Set of Transmission Eigenvalues , 2010, SIAM J. Math. Anal..

[11]  Petri Ola,et al.  Transmission Eigenvalues for Elliptic Operators , 2010, SIAM J. Math. Anal..

[12]  E. Lakshtanov,et al.  Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem , 2012, 1212.6785.

[13]  H. Haddar,et al.  Transmission Eigenvalues in Inverse Scattering Theory , 2012 .

[14]  John Sylvester,et al.  Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators , 2011, SIAM J. Math. Anal..

[15]  John Sylvester,et al.  Transmission eigenvalues for degenerate and singular cases , 2012 .

[16]  Evgeny Lakshtanov,et al.  Remarks on interior transmission eigenvalues, Weyl formula and branching billiards , 2011, 1112.0891.

[17]  Ha Pham,et al.  Weyl asymptotics of the transmission eigenvalues for a constant index of refraction , 2013, 1309.3616.

[18]  Luc Robbiano,et al.  Spectral analysis of the interior transmission eigenvalue problem , 2013, 1302.4851.

[19]  Transmission Eigenvalues in One Dimension , 2013, 1305.0733.

[20]  L. Päivärinta,et al.  Completeness of generalized transmission eigenstates , 2013 .

[21]  Counting function for interior transmission eigenvalues , 2013, 1310.6273.

[22]  Weyl Type Bound on Positive Interior Transmission Eigenvalues , 2014 .

[23]  Melvin Faierman.  The Interior Transmission Problem: Spectral Theory , 2014, SIAM J. Math. Anal..

[24]  Georgi Vodev,et al.  Asymptotics of the number of the interior transmission eigenvalues , 2014, 1403.3949.

[25]  Georgi Vodev,et al.  Transmission Eigenvalue-Free Regions , 2014, 1401.1627.