Reflection Probabilities of One-Dimensional Schrödinger Operators and Scattering Theory

Abstract

The dynamic reflection probability of Davies and Simon (Commun Math Phys 63(3):277–301, 1978) and the spectral reflection probability of Gesztesy et al. (Diff Integral Eqs 10(3):521–546, 1997) and Gesztesy and Simon (Helv Phys Acta 70:66–71, 1997) for a one-dimensional Schrödinger operator $H = - \Delta + V$ are characterized in terms of the scattering theory of the pair $(H, H_\infty )$ where $H_\infty$ is the operator obtained by decoupling the left and right half-lines $\mathbb {R}_{\le 0}$ and $\mathbb {R}_{\ge 0}$. An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of Breuer et al. (Commun Math Phys 295(2):531–550, 2010). This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black-box model and follows a strategy parallel to Jakšić (Commun Math Phys 332:827–838, 2014).