Scattering for Schrödinger operators with potentials concentrated near a subspace

Black, Adam; Malinovitch, Tal

doi:10.1090/tran/8854

Scattering for Schrödinger operators with potentials concentrated near a subspace
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by Adam Black and Tal Malinovitch;

Trans. Amer. Math. Soc. 376 (2023), 2525-2555

DOI: https://doi.org/10.1090/tran/8854

Published electronically: January 23, 2023

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Abstract:

We study the scattering properties of Schrödinger operators with bounded potentials concentrated near a subspace of $\mathbb {R}^d$. For such operators, we show the existence of scattering states and characterize their orthogonal complement as a set of surface states, which consists of states that are confined to the subspace (such as pure point states) and states that escape it at a sublinear rate, in a suitable sense. We provide examples of surface states for different systems including those that propagate along the subspace and those that escape the subspace arbitrarily slowly. Our proof uses a novel interpretation of the Enss method [Comm. Math. Phys. 61 (1978), pp. 285–291] in order to obtain a dynamical characterization of the orthogonal complement of the scattering states.

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