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Quantum ergodicity for Dirichlet-truncated operators on $\mathbb{Z}^d$

Hongyi Cao, Shengquan Xiang

TL;DR

The paper addresses quantum ergodicity for Dirichlet-truncated operators on $\mathbb Z^d$, proving QE for the Dirichlet-truncated adjacency and establishing partial QE for Dirichlet Schrödinger operators with small-period lengths. It introduces two complementary proofs: a four-step quantum-variance method with explicit matrix calculations in a Dirichlet-adapted sine basis, and an eigenfunction correspondence that embeds Dirichlet eigenfunctions into periodic ones on a larger cube to transfer QE results. The results delineate a clear contrast with periodic boundary conditions, and extend QE to finite-range observables, thereby partially answering an open question by McKenzie and Sabri. The work advances the understanding of delocalization and ergodicity in lattice systems under Dirichlet boundaries and suggests pathways to connect Dirichlet and periodic spectral theories through eigenfunction correspondences.

Abstract

In this paper, we prove quantum ergodicity (a form of delocalization for eigenfunctions) for the Dirichlet truncations of the adjacency matrix on $\mathbb{Z}^d$. We also extend the result to the cases of finite range observables and periodic Schrödinger operators with periods of length at most two. This work partially answers a question asked by McKenzie and Sabri (Comm. Math. Phys. 403(3), 1477--1509(2023)).

Quantum ergodicity for Dirichlet-truncated operators on $\mathbb{Z}^d$

TL;DR

The paper addresses quantum ergodicity for Dirichlet-truncated operators on , proving QE for the Dirichlet-truncated adjacency and establishing partial QE for Dirichlet Schrödinger operators with small-period lengths. It introduces two complementary proofs: a four-step quantum-variance method with explicit matrix calculations in a Dirichlet-adapted sine basis, and an eigenfunction correspondence that embeds Dirichlet eigenfunctions into periodic ones on a larger cube to transfer QE results. The results delineate a clear contrast with periodic boundary conditions, and extend QE to finite-range observables, thereby partially answering an open question by McKenzie and Sabri. The work advances the understanding of delocalization and ergodicity in lattice systems under Dirichlet boundaries and suggests pathways to connect Dirichlet and periodic spectral theories through eigenfunction correspondences.

Abstract

In this paper, we prove quantum ergodicity (a form of delocalization for eigenfunctions) for the Dirichlet truncations of the adjacency matrix on . We also extend the result to the cases of finite range observables and periodic Schrödinger operators with periods of length at most two. This work partially answers a question asked by McKenzie and Sabri (Comm. Math. Phys. 403(3), 1477--1509(2023)).
Paper Structure (16 sections, 9 theorems, 102 equations, 1 figure)

This paper contains 16 sections, 9 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Comparison between periodic conditions and Dirichlet conditions
  4. Main ideas and new ingredients
  5. Explicit matrix calculations and Fourier coefficient estimation
  6. Eigenfunction correspondence
  7. Quantum ergodicity for the adjacency matrix on $\mathbb Z^d$ with Dirichlet conditions
  8. Eigenfunction correspondence
  9. Established results in the periodic conditions case
  10. The injective map $\mathfrak{e}$ and eigenfunction correspondence
  11. Proof of Theorem \ref{['thm']}
  12. Proof of Theorem \ref{['thm.']}
  13. Proof of Proposition \ref{['app']} and Theorem \ref{['ps']}
  14. Further comments
  15. About the eigenfunction correlators in Theorem \ref{['thm.']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{\psi_N^{(j)}\}_{j=1}^{N^d}$ be an orthonormal basis of $\ell^2(\Gamma_N)$ consisting of eigenfunctions of $\mathcal{A}_{\Gamma_N}$ and let $a_N: \Gamma_N\times \Gamma_N\to \mathbb C$ be a sequence of diagonal observables satifying $\|a_N\|_\infty:=\sup_{\bm{x}\in\Gamma_N}|a_N(\bm{x},\bm{x})|\ where $\langle\cdot,\cdot \rangle$ is the inner product so that $\langle \psi_N^{(j)}, a_N \psi_N^{

Figures (1)

  • Figure 1: A description of the map $\mathfrak{e}$ for $d=2,N=3$: $\mathfrak{e}\psi$ takes $0$ at the blue points, the same sign as $\psi$ at the black points, and the positive sign to $\psi$ at the red points.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.4
  • Theorem 1.5
  • Definition 3.1
  • Theorem 3.2: MS23, diagonal observables
  • Theorem 3.3: MS23, finite range observables
  • Definition 3.5
  • Proposition 3.6
  • proof : Proof of Theorem \ref{['thm']}
  • ...and 7 more