Sharp polynomial bounds on the number of scattering poles of radial potentials. (English) Zbl 0681.47002

Zworski, Maciej

J. Funct. Anal. 82, No. 2, 370-403 (1989).

Author’s summary: It is shown that for scattering by a radially symmetric potential in \({\mathbb{R}}^ N\), N odd, the number of poles in a disk of radius r satisfies an estimate \[ n(r)\leq C_ N(r+1)^ N. \] This bound is sharp as shown by the special case of potentials nonvanishing at the boundary, where \(n(r)=K_ Na^ Nr^ N(1+o(1)),\) a being the diameter of the support.

Reviewer: J.Appell

MSC:

47A40	Scattering theory of linear operators
34L99	Ordinary differential operators
35P25	Scattering theory for PDEs
35J10	Schrödinger operator, Schrödinger equation

Keywords:

scattering by a radially symmetric potential

References:

[1]	Faddeev, L. D., The inverse problem in the quantum theory of scattering, J. Math. Phys., 4, 72-104 (1961) · Zbl 0112.45101
[2]	Intissar, A., A polynomial bound on the number of scattering poles for a potential in even dimensional space \(\textbf{R}^n \), Comm. Partial Differential Equations, 11, No. 4, 367-396 (1986) · Zbl 0607.35069
[3]	Lax, P. D.; Phillips, R. S., Scattering Theory (1967), Academic Press: Academic Press Orlando, FL · Zbl 0214.12002
[4]	Lax, P. D.; Phillips, R. S., Decaying modes for the wave equation in the exterior of an obstacle, Comm. Pure Appl. Math., 22, 737-787 (1969) · Zbl 0181.38201
[5]	Melin, A., Operator methods for inverse scattering on the real line, Comm. Partial Differential Equations, 10, No. 7, 677-766 (1985) · Zbl 0585.35077
[6]	Melin, A., Intertwining methods for Schrödinger operator in \(\textbf{R}^n (1986)\), preprint
[7]	Melrose, R. B., Scattering theory and the trace of the wave group, J. Funct. Anal., 45, 29-40 (1982) · Zbl 0525.47007
[8]	Melrose, R. B., Polynomial bounds on the number of scattering poles, J. Funct. Anal., 53, 287-303 (1983) · Zbl 0535.35067
[9]	Melrose, R. B., Growth estimates for the poles in potential scattering (1984), unpublished · Zbl 0554.35085
[10]	Melrose, R. B., Polynomial bounds on the distribution of poles in scattering by an obstacle, (Journées “Equations aux Dérivées Partielle” (1984)), Saint-Jean-des-Montes · Zbl 0435.35066
[11]	Melrose, R. B.; Taylor, M., Boundary problems for wave equations with grazing and gliding rays (1986), preprint
[12]	Newton, R., Analytic properties of radial wave functions, J. Math. Phys., 1, 319-405 (1960) · Zbl 0090.19303
[13]	Nussenzweig, H. M., The poles of \(S\)-matrix of a rectangular potential well or barrier, Nuclear Phys., 11, 499-521 (1959)
[14]	Olver, F. W.J, The asymptotic expansions of Bessel functions of large order, Philos. Trans. Roy. Soc. London Ser. A, 47, 328-368 (1954) · Zbl 0070.30801
[15]	Regge, T., Analytic properties of the scattering matrix, Nuovo Cimento, 8, No. 5, 671-681 (1958) · Zbl 0080.41903
[16]	Shenk, N.; Thoe, D., Resonant states and the poles of the scattering matrix for perturbations of −Δ, J. Math. Anal. AAppl., 37, 467-491 (1972) · Zbl 0229.35072
[17]	Wei, M.; Majda, G.; Strauss, W., Numerical computation of the scattering frequencies for the acoustic wave equation (1986), preprint
[18]	Titchmarsh, E. C., The Theory of Functions (1968), Oxford Univ. Press: Oxford Univ. Press London/New York · Zbl 0005.21004
[19]	Watson, G. N., A Treatise on the Theory of Bessel Functions (1944), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0063.08184
[20]	Zworski, M., Distribution of poles for scattering on the real line, J. Funct. Anal., 73, 277-296 (1987) · Zbl 0662.34033