Distribution of poles for scattering on the real line. (English) Zbl 0662.34033

Zworski, Maciej

J. Funct. Anal. 73, 277-296 (1987).

The density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential. In the case of a potential with finite singularities at the endpoints of the support, asymptotic formulae for the poles are given, while in the \(C^{\infty}_0\) case, an example of a potential with infinitely many scattering poles on \({\mathbb{R}}\) is constructed. The scattering amplitude of a compactly supported potential is also characterized.

Reviewer: P. Bolley (Nantes)

MSC:

34L25

Scattering theory, inverse scattering involving ordinary differential operators

Keywords:

density of scattering poles; potential

References:

[1]	Cartwright, M. L., The zeros of certain integral functions II, Quart. J. Math. (Oxford), 2, 113-129 (1931) · JFM 57.0361.03
[2]	Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., 32, 121-251 (1979) · Zbl 0388.34005
[3]	Hardy, G. H., On the zeros of certain classes of integral Taylor series, II, (Proc. London Math. Soc., 2 (1904)), 411-431 · JFM 36.0473.03
[4]	Hörmander, L., The Analysis of Linear Partial Differential Operators, I, II (1983), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg · Zbl 0521.35002
[5]	Melin, A., Operator methods for inverse scattering on the real line, Comm. Partial Differential Equations, 10, 7, 677-766 (1985) · Zbl 0585.35077
[6]	Melrose, R., Polynomial bounds on the number of scattering poles, J. Funct. Anal., 53, 3, 287-303 (1983) · Zbl 0535.35067
[7]	Regge, T., Analytic properties of the scattering matrix, Nuovo Cimento, 8, 5, 671-679 (1958) · Zbl 0080.41903
[8]	Titchmarsh, E. C., The zeros of certain integral functions, (Proc. London Math. Soc., 25 (1926)), 283-302 · JFM 52.0334.03