Low-energy behaviour of the scattering matrix for the Schrödinger equation on the line. (English) Zbl 0669.34030

Klaus, Martin

Inverse Probl. 4, No. 2, 505-512 (1988).

For potentials whose first absolute moment exists we prove continuity of the scattering matrix at zero energy. As a result we obtain Levinson’s theorem. We also study the low-energy asymptotics of the transmission and reflection coefficients for potentials with a behaviour of the type \(x^{-2-\epsilon}\) (respectively \((-x)^{-2-\delta})\) as \(x\to \infty\) (respectively \(x\to -\infty)\), where \(0<\epsilon\), \(\delta <1\).

MSC:

34L99

Ordinary differential operators

Keywords:

Levinson’s theorem; transmission; reflection coefficients; potentials