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Scattering for Schrödinger operators with potentials concentrated near a subspace

Adam Black, Tal Malinovitch

TL;DR

This work extends scattering theory for Schrödinger operators to potentials concentrated near a subspace by proving the existence of wave operators and establishing asymptotic completeness in the presence of surface states. The authors introduce a novel Enss-based framework tailored to surface scattering, employing Davies phase-space observables $P_\delta(E)$ to decompose dynamics into incoming, outgoing, and surface components and to analyze transverse motion along the surface. They characterize the surface subspace $\mathcal{H}_{\mathrm{sur}}$ dynamically and demonstrate that $\mathcal{H}=\mathcal{H}_{\mathrm{sur}}\oplus\mathrm{Ran}(\Omega^-)$, with $\mathcal{H}_{\mathrm{sur}}=\tilde{\mathcal{H}}_{\mathrm{sur}}$ shown via non-stationary phase arguments. The paper also provides concrete examples, including partially periodic and transient surface states, and discusses perturbative and random-surface scenarios, highlighting the richness of surface phenomena and their impact on asymptotic completeness in anisotropic quantum systems.

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 in order to obtain a dynamical characterisation of the orthogonal complement of the scattering states.

Scattering for Schrödinger operators with potentials concentrated near a subspace

TL;DR

This work extends scattering theory for Schrödinger operators to potentials concentrated near a subspace by proving the existence of wave operators and establishing asymptotic completeness in the presence of surface states. The authors introduce a novel Enss-based framework tailored to surface scattering, employing Davies phase-space observables to decompose dynamics into incoming, outgoing, and surface components and to analyze transverse motion along the surface. They characterize the surface subspace dynamically and demonstrate that , with shown via non-stationary phase arguments. The paper also provides concrete examples, including partially periodic and transient surface states, and discusses perturbative and random-surface scenarios, highlighting the richness of surface phenomena and their impact on asymptotic completeness in anisotropic quantum systems.

Abstract

We study the scattering properties of Schrödinger operators with bounded potentials concentrated near a subspace of . 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 in order to obtain a dynamical characterisation of the orthogonal complement of the scattering states.
Paper Structure (21 sections, 26 theorems, 222 equations)

This paper contains 21 sections, 26 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Motivation and prior work
  3. Model and results
  4. Methodology: the Enss Method of scattering
  5. Definitions and Results
  6. Notation and Conventions
  7. Definition of the auxiliary surface subspace
  8. Existence of the Wave Operators
  9. Enss Decomposition
  10. Proof of part \ref{['Complete']} of Theorem \ref{['thms']}: Asymptotic completeness
  11. Step 1: The Span of $\mathop{\mathrm{Ran}}\nolimits\Omega^-$ and $\tilde{\mathcal{H}}_\mathrm{sur}$
  12. Step 2: The Intersection of $\mathop{\mathrm{Ran}}\nolimits\Omega^-$ and $\tilde{\mathcal{H}}_\mathrm{sur}$
  13. Step 3: $\tilde{\mathcal{H}}_{\mathrm{sur}}=\mathcal{H}_{\mathrm{sur}}$
  14. Examples
  15. Surface States in $\sigma_{\mathrm{c}}(H)$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Theorems & Definitions (61)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 3.1
  • proof : Proof of part \ref{['existenceTheorem']} of Theorem \ref{['thms']}
  • Theorem 4.1
  • proof : Proof of Theorem \ref{['thm:decomposition']}
  • Lemma 4.2
  • proof
  • ...and 51 more