Modulus of continuity eigenvalue bounds for homogeneous graphs and convex subgraphs with applications to quantum Hamiltonians

TL;DR

This paper introduces a novel technique using modulus of continuity estimates to bound the spectral gap of graph Laplacians and stoquastic Hamiltonians, linking PDE methods with quantum and combinatorial spectral analysis.

Contribution

It adapts PDE-based modulus of continuity methods to derive eigenvalue bounds for graph Laplacians and quantum Hamiltonians, providing new insights into their spectral properties.

Findings

Bound the spectral gap of weighted Laplacians and Hamiltonians

Establish connections between PDE techniques and spectral graph theory

Apply recent PDE advances to quantum Hamiltonian analysis

Abstract

We adapt modulus of continuity estimates to the study of spectra of combinatorial graph Laplacians, as well as the Dirichlet spectra of certain weighted Laplacians. The latter case is equivalent to stoquastic Hamiltonians and is of current interest in both condensed matter physics and quantum computing. In particular, we introduce a new technique which bounds the spectral gap of such Laplacians (Hamiltonians) by studying the limiting behavior of the oscillations of their eigenvectors when introduced into the heat equation. Our approach is based on recent advances in the PDE literature, which include a proof of the fundamental gap theorem by Andrews and Clutterbuck.