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

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
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