Reflection probabilities of one-dimensional Schrödinger operators and scattering theory

Abstract : The dynamic reflection probability and the spectral reflection probability for a one-dimensional Schroedinger 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 $\mathbb{R}_{\leq 0}$ and $\mathbb{R}_{\geq 0}$. 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, J., E. Ryckman, and B. Simon (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 the Jacobi case.
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Benjamin Landon, Jane Panangaden, Annalisa Panati, Justine Zwicker. Reflection probabilities of one-dimensional Schrödinger operators and scattering theory. Annales Henri Poincaré, Springer Verlag, 2017, 18 (6), pp.2075-2085. ⟨10.1007/s00023-016-0543-0⟩. ⟨hal-01263094⟩

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