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Directional Ballistic Transport in Quantum Waveguides

Adam Black, David Damanik, Peter Kuchment, Tal Malinovitch, Giorgio Young

TL;DR

The paper studies Schrödinger operators on R^{n+m} with potentials that are compact in the x-directions and periodic in the y-directions, producing surface states localized near the potential. It develops an analytic Floquet framework based on a two-sided Dirichlet-to-Neumann map to convert the surface-state eigenvalue problem into a Fredholm problem on a bounded domain, enabling analytic dependence of surface-band energies on quasimomentum k. The authors prove that surface states exhibit directional ballistic transport: ballistic motion along the periodic directions and confinement in the transverse directions, with a decomposition of the surface subspace into analytically varying bands whose energies are non-constant almost everywhere. This leads to a rigorous mechanism for ballistic transport in higher-dimensional open waveguides and confirms purely absolutely continuous spectrum on the surface subspace, extending prior discrete results to the continuum and providing a robust, operator-theoretic approach to transport phenomena in partially periodic media.

Abstract

We study the transport properties of Schrödinger operators on $\mathbb{R}^d$ with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce surface states that are weakly confined near the support of the potential. We show that a natural set of surface states exhibits directional ballistic transport, characterized by ballistic transport in the periodic directions and its absence in the others. To prove this, we develop a Floquet theory that captures the analytic variation of surface states. The main idea consists of reformulating the eigenvalue problem for surface states as a Fredholm problem via the Dirichlet-to-Neumann map.

Directional Ballistic Transport in Quantum Waveguides

TL;DR

The paper studies Schrödinger operators on R^{n+m} with potentials that are compact in the x-directions and periodic in the y-directions, producing surface states localized near the potential. It develops an analytic Floquet framework based on a two-sided Dirichlet-to-Neumann map to convert the surface-state eigenvalue problem into a Fredholm problem on a bounded domain, enabling analytic dependence of surface-band energies on quasimomentum k. The authors prove that surface states exhibit directional ballistic transport: ballistic motion along the periodic directions and confinement in the transverse directions, with a decomposition of the surface subspace into analytically varying bands whose energies are non-constant almost everywhere. This leads to a rigorous mechanism for ballistic transport in higher-dimensional open waveguides and confirms purely absolutely continuous spectrum on the surface subspace, extending prior discrete results to the continuum and providing a robust, operator-theoretic approach to transport phenomena in partially periodic media.

Abstract

We study the transport properties of Schrödinger operators on with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce surface states that are weakly confined near the support of the potential. We show that a natural set of surface states exhibits directional ballistic transport, characterized by ballistic transport in the periodic directions and its absence in the others. To prove this, we develop a Floquet theory that captures the analytic variation of surface states. The main idea consists of reformulating the eigenvalue problem for surface states as a Fredholm problem via the Dirichlet-to-Neumann map.
Paper Structure (16 sections, 27 theorems, 142 equations)

This paper contains 16 sections, 27 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Overview of the proof
  5. Preliminaries
  6. Floquet Theory
  7. Domain division and function spaces
  8. Construction of the Dirichlet-to-Neumann map
  9. The interior boundary value problem
  10. The exterior boundary value problem
  11. The two-sided Dirichlet-to-Neumann map
  12. Fredholm property of the Dirichlet-to-Neumann map
  13. Proofs of the main theorems
  14. Analytical variation of the kernel of left semi-Fredholm operators
  15. Non-constancy of the energies
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $H$ be a Schrödinger operator with potential $V$ satisfying eq:VasumpX and eq:VasumpY. Then every state $\psi \in D(Q)\cap H^2(\mathbb{R}^{n+m})\cap \mathcal{H}_{\mathrm{sur}}\setminus\{0\}$ exhibits directional ballistic transport under the evolution of $H$.

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 41 more