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.

%B Journal of Mathematical Analysis and Applications %V 452 %P 1269-1290 %8 2017/03/03 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022247X1730272X %N 2 %R 10.1016/j.jmaa.2017.03.030