×

Lower bounds on the number of scattering poles. (English) Zbl 0784.35070

The authors provide several examples for which they can give lower bounds on the number of scattering poles inside a ball of radius \(R\), with \(R\) large enough. Their examples include scattering by obstacles, perturbations of the Euclidean metric, and hyperbolic manifolds. In each case, they obtain a lower bound growing polynomially as \(R\) tends to infinity.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P25 Scattering theory for PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bardos C., La relotton de Porsson pour l’iquation des ondes dans un ouvertnon bornd 7 pp 905– (1982)
[2] Bardos C., Scattering frequencies and Geurey 3 singularities 90 pp 77– (1987)
[3] Duistermaat J.J, The spectrum of positrue elltptic operators and periodic bicharac- teristics 29 pp 39– (1975)
[4] Gérard C., Asymptotique des pÔIes de la matrice de scattering pour deuz obstacles strictement convezes 116 (1988)
[5] Guillemin V.W., The Poisson summation formula for manifolds with boundaries 32 pp 204– (1979) · Zbl 0421.35082
[6] Hörmander L., The Analysis of Linear Partial Differential Operators (1963)
[7] Ikawa M., On the distribution of poles ofthe scattering matrix for two strictly conuez obstaclesobstacles. Iiokkaido Math. J. 12 pp 343– (1988)
[8] Ikawa M., On the existence of poles of the scattering matriz for several conuez bodies. Proc. Japan. Acad. 64 pp 91– (1988)
[9] Lax P., Scatterrng theory (1967)
[10] Melrose R. B., Scattering theory and the trace ofthe wave group 45 pp 29– (1982) · Zbl 0525.47007
[11] Patterson S. J., The Selberg zeta-function of a Kleinian group pp 409– (1987)
[12] Perry P. A., The Selberg Zeta funciion and a local trace formula for Kleinian groups. J 410 pp 116– (1990) · Zbl 0697.10027
[13] Perry P. A., The Selberg zeta function and scattering poles for Kleinran groups 24 pp 327– (1991) · Zbl 0723.11028
[14] Sjöstrand J., Complez scaling and the distribution of scattering poles. 4 pp 729– (1991) · Zbl 0752.35046
[15] Sjostrand J., Comm.P.D.E. to appear.
[16] Vodev G., Sharp polynomial bounds on the number of scattering poles for metric perturbations of Laplacian rn Rn n, odd 291 pp 39– (1991) · Zbl 0754.35105
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.