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Modulus of continuity eigenvalue bounds for homogeneous graphs and convex subgraphs with applications to quantum Hamiltonians. (English) Zbl 1365.81053

Summary: 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 stochastic 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 solutions 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.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
05C30 Enumeration in graph theory
35K05 Heat equation
82D20 Statistical mechanics of solids
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References:

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