A complete treatment of low-energy scattering in one dimension. (English) Zbl 0567.47008

Bollé, D.; Gesztesy, F.; Wilk, S. F. J.

J. Oper. Theory 13, 3-32 (1985).

The authors present a systematic analysis of low-energy scattering for Schrödinger Hamiltonians, H, on the entire real line, taking into account explicitly the possibility of zero-energy resonances of H. The possible occurrence of these resonances leads to a classification of essentially two cases i.e. H has no zero-energy resonances \((=\) generic case) and H has a zero-energy resonance of multiplicity 1. The potential conditions are \[ (1)\quad \int_{{\mathbb{R}}}dx(1+| x|^ 2)| V(x)| <\infty \quad and\quad (2)\quad \int_{{\mathbb{R}}}dx V(x)\neq 0. \] For these cases the low-energy behavior of the transition operator T(k) is described. In particular, for potentials satisfying conditions (1), (2) and (3) \(\int_{{\mathbb{R}}}dx[\exp a| x|]| V(x)| <\infty,\) \(a>0\), recursion relations are established for the coefficients in the Taylor expansions for T(k) (generic case) or Laurent expansions for T(k) (other cases). Also Taylor expansions for the reflection and transmission coefficients are presented in all cases. Finally, two sets of trace relations involving the continuous and point spectrum of H are proved. As a special case Levinson’s theorem for scattering on the line is obtained. Its structure completely changes in comparison with three dimensions.
If one is interested in asymptotic expansions instead of analytic ones, it is shown how the exponential fall off conditions on the potential can be relaxed. An extension of this work replacing condition (2) by \(\int_{{\mathbb{R}}}dx V(x)=0\), by D. Bollé, F. Gesztesy and M. Klaus will appear in J. Math. Anal. Appl. (1985).

MSC:

47A40	Scattering theory of linear operators
81U05	\(2\)-body potential quantum scattering theory
34L99	Ordinary differential operators
47B06	Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
46N99	Miscellaneous applications of functional analysis

Keywords:

low-energy scattering for Schrödinger Hamiltonians; zero-energy resonances; low-energy behavior of the transition operator; Taylor expansions; Laurent expansions; reflection and transmission coefficients; trace relations involving the continuous and point spectrum; Levinson’s theorem for scattering on the line; asymptotic expansions