Differential and Integral Equations

One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics

F. Gesztesy, R. Nowell, and W. Pötz

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We provide a general framework of stationary scattering theory for one-dimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reflectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Shrödinger operator. Moreover, we prove that single (Crum-Darboux) and double commutation methods to insert eigenvalues into spectral gaps of a given background Shrödinger operator produce reflectionless potentials (i.e., solitons) if and only if the background potential is reflectionless. Possible applications of our formalism include impurity (defect) scattering in (half-) crystals and charge transport in mesoscopic quantum-interference devices.

Article information

Differential Integral Equations, Volume 10, Number 3 (1997), 521-546.

First available in Project Euclid: 2 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81U05: $2$-body potential scattering theory [See also 34E20 for WKB methods]
Secondary: 34B20: Weyl theory and its generalizations 34L25: Scattering theory, inverse scattering 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47)


Gesztesy, F.; Nowell, R.; Pötz, W. One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics. Differential Integral Equations 10 (1997), no. 3, 521--546. https://projecteuclid.org/euclid.die/1367525666

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