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

TL;DR

This paper establishes quantum ergodicity for Dirichlet-truncated adjacency matrices on integer lattices, extending to finite range observables and certain periodic Schrödinger operators, thus advancing understanding of eigenfunction delocalization.

Contribution

It proves quantum ergodicity for Dirichlet truncations on ^d and extends results to finite range observables and specific periodic Schrödinger operators, addressing a recent open question.

Findings

Quantum ergodicity holds for Dirichlet-truncated adjacency matrices on ^d.

Results extend to finite range observables.

Includes periodic Schrödinger operators with small periods.

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 Schr\"odinger 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)).