One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics
The authors establish a general framework of stationary scattering theory of one-dimensional quantum systems with nontrivial spatial asymptotics. The principal method used consists in combination of the Weyl-Titchmarsh theory of second-order differential operators generated by the differential expression \[ \tau =-{d^2\over dt^2}+ V(t),\quad t\in \mathbb{R}, \] with explicit spectral representation in terms of the generalized eigenfunction expansion for such operators. Using this method, the explicit formulas for the so-called transmission and reflection coefficients \(T^{\pm }(\lambda)\), \(R^{\pm }(\lambda)\) via Weyl \(m\)-functions and normalized real eigenfunctions are obtained as well as the detailed description of the connection between scattering data and spectral representation of the underlying Schrödinger operators. It is also shown that the obtained results have a broad range of application in various (physical) situations, like charge transport in quasi-one-dimensional mesoscopic solid state structure.
scientific article; zbMATH DE number 1153421