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We adapt modulus of continuity estimates to the study of spectra ofcombinatorial graph Laplacians, as well as the Dirichlet spectra of certainweighted Laplacians. The latter case is equivalent to stoquastic Hamiltoniansand is of current interest in both condensed matter physics and quantumcomputing. In particular, we introduce a new technique which bounds thespectral gap of such Laplacians (Hamiltonians) by studying the limitingbehavior of the oscillations of their eigenvectors when introduced into theheat equation. Our approach is based on recent advances in the PDE literature,which include a proof of the fundamental gap theorem by Andrews andClutterbuck.

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