×

Sharp Weyl law for signed counting function of positive interior transmission eigenvalues. (English) Zbl 1331.35252

The main result of the paper (Theorem 1.1) is a remarkable Weyl-type law regarding the positive interior transmission eigenvalues. It, also, presents some interesting properties of the eigenvalues of the corresponding scattering matrix. Overall it is an informative and well-written article.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
47A53 (Semi-) Fredholm operators; index theories
35J57 Boundary value problems for second-order elliptic systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. F. Atiyah, V. Patodi, and I. M. Singer, {\it Spectral asymmetry and Riemannian geometry} III, Math. Proc. Cambridge Philos. Soc., 79 (1976), pp. 71-99. · Zbl 0325.58015
[2] M. Birman and D. Yafaev, {\it Spectral properties of the scattering matrix,} St. Petersburg Math. J., 4 (1993), pp. 1-27. · Zbl 0819.47009
[3] E. Blasten and L. Paivarinta, {\it Completeness of generalized transmission eigenstates,} Inverse Problems, 29 (2013), 104002. · Zbl 1294.35042
[4] E. Blasten, L. Paivarinta, and J. Sylvester, {\it Do corners always scatter?,} Comm. Math. Phys., 331 (2014), pp. 725-753. · Zbl 1298.35214
[5] P. Bleher, {\it Operators that depend meromorphically on a parameter}, Vestnik Moskov. Univ. Ser. I Mat. Meh., 24 (1969), pp. 30-36 (in Russian).
[6] D. Colton and L. Paivarinta, {\it Transmission eigenvalues and a problem of Hans Lewy}, J. Comput. Appl. Math., 117 (2000), pp. 91-104. · Zbl 0957.65093
[7] F. Cakoni and H. Haddar, {\it Transmission eigenvalues in inverse scattering theory, } in Inverse Problems and Applications: Inside Out II, Math. Sci. Res. Inst. Publ. 60, Cambridge University Press, Cambridge, UK, 2012, pp. 529-580. · Zbl 1316.35297
[8] D. Colton and P. Monk, Quart. J. Mech. Appl. Math., 41 (1988), pp. 97-125.
[9] M. Dimassi and V. Petkov, {\it Upper Bound for the Counting Function of Interior Transmission Eigenvalues}, preprint, arXiv:1308.2594, 2013.
[10] E. Doron and U. Smilansky, {\it Semiclassical quantization of chaotic billiards: A scattering theory approach}, Nonlinearity, 5 (1992), pp. 1055-1084. · Zbl 0770.58043
[11] J. P. Eckmann and C.-A. Pillet, {\it Spectral duality for planar billiards,} Comm. Math. Phys., 170 (1995), pp. 283-313. · Zbl 0834.35096
[12] J. P. Eckmann and C.-A. Pillet, {\it Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains}, Helv. Phys. Acta, 70 (1997), pp. 44-65. · Zbl 0871.35070
[13] M. Faierman, {\it The interior transmission problem: Spectial theory}, SIAM J. Math. Anal., 46 (2014), pp. 803-819. · Zbl 1294.35049
[14] L. Friedlander, {\it Some inequalities between Dirichlet and Neumann eigenvalues, } Arch. Rational Mech. Anal., 116 (1991), pp. 153-160. · Zbl 0789.35124
[15] I. Gohberg, P. Lancaster, and L. Rodman, {\it Indefinite Linear Algebra and Applications}, Birkhäuser, Basel, 2005. · Zbl 1084.15005
[16] A. Jensen and T. Kato, {\it Asymptotic behavior of the scattering phase for exterior domains}, Comm. Partial Differential Equations, 3 (1978), pp. 1165-1195. · Zbl 0419.35067
[17] T. Kato, {\it Monotonicity theorems in scattering theory}, Hadronic J., 1 (1978), pp. 134-154. · Zbl 0426.47004
[18] A. Kirsch, {\it The denseness of the far field patterns for the transmission problem}, IMA J. Appl. Math., 37 (1986), pp. 213-223. · Zbl 0652.35104
[19] A. Kirsch and A. Lechleiter, {\it The inside-outside duality for scattering problems by inhomogeneous media,} Inverse Problems, 29 (2013), 104011. · Zbl 1285.35058
[20] E. Lakshtanov and B. Vainberg, {\it Ellipticity in the interior transmission problem in anisotropic media}, SIAM J. Math. Anal., 44 (2012), pp. 1165-1174. · Zbl 1245.35126
[21] E. Lakshtanov and B. Vainberg, {\it Remarks on interior transmission eigenvalues, Weyl formula and branching billiards,} J. Phys. A, 45 (2012), pp. 125-202. · Zbl 1245.81290
[22] E. Lakshtanov and B. Vainberg, {\it Bounds on positive interior transmission eigenvalues, } Inverse Problems, 28 (2012), 105005. · Zbl 1256.35036
[23] E. Lakshtanov and B. Vainberg, {\it Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem,} Inverse Problems, 29 (2013), 104003. · Zbl 1285.35059
[24] E. Lakshtanov and B. Vainberg, {\it Weyl type bound on positive interior transmission eigenvalues,} Comm. Partial Differential Equations, 39 (2014), pp. 1729-1740. · Zbl 1301.35078
[25] Y. J. Leung and D. Colton, {\it Complex transmission eigenvalues for spherically stratified media,} Inverse Problems, 28 (2012), 075005. · Zbl 1260.47012
[26] A. Majda and J. V. Ralston, {\it An analogue of Weyl’s theorem for unbounded domains.} I, Duke Math. J., 45 (1978), pp. 183-196. · Zbl 0408.35069
[27] V. Petkov and G. Vodev, {\it Asymptotics of the Number of the Interior Transmission Eigenvalues}, preprint, arXiv:1403.3949, 2014. · Zbl 1458.35291
[28] H. Pham and P. Stefanov,{\it Weyl asymptotics of the transmission eigenvalues for a constant index of refraction,} Inverse Probl. Imaging, 8 (2014), pp. 795-810. · Zbl 1432.35155
[29] A. Pushnitski,{\it The spectral shift function and the invariance principle}, J. Funct. Anal., 183 (2001), pp. 269-320. · Zbl 0998.47003
[30] M. Reed and B. Simon, {\it Methods of Modern Mathematical Physics.} IV, Academic Press, New York, London, 1978. · Zbl 0401.47001
[31] D. Robert, {\it Asymptotique de la phase de diffusion a haute energie pour des perturbations du second ordre du Laplacien,} Ann. Sci. École Norm. Sup. (4), 25 (1992), pp. 107-134. · Zbl 0801.35100
[32] L. Robbiano, {\it Spectral analysis on interior transmission eigenvalues}, Inverse Problems, 29 (2013), 104001. · Zbl 1296.35105
[33] B. P. Rynne and B. D. Sleeman, {\it The interior transmission problem and inverse scattering from inhomogeneous media}, SIAM J. Math. Anal., 22 (1991), pp. 1755-1762. · Zbl 0733.76065
[34] Yu. Safarov and D. Vassiliev, {\it The Asymptotic Distribution of Eigenvalues of Partial Differential Operators}, American Mathematical Society, Providence, RI, 1997. · Zbl 0870.35003
[35] O. Safronov, {\it The discrete spectrum in gaps of the continuous spectrum for indefinite-sign perturbations with a large coupling constant}, St. Petersburg Math. J., 8 (1997), pp. 307-331.
[36] O. Safronov, {\it Spectral shift function in the large coupling constant limit}, J. Funct. Anal., 182 (2001), pp. 151-169. · Zbl 0993.47013
[37] U. Smilansky, {\it Semiclassical quantization of chaotic billiards: A scattering approach}, in Mesoscopic Quantum Physics (Proc. of the Les Houches Summer School on Mesoscopic Quantum Physics), E. Akkermans, G. Montambaux, J.- L. Pichard, and J. Zinn-Justin, eds., Elsevier Science, Amsterdam, 1995, pp. 373-434.
[38] J. Sylvester, {\it Discreteness of transmission eigenvalues via upper triangular compact operators}, SIAM J. Math. Anal., 44 (2012), pp. 341-354. · Zbl 1238.81172
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.