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Localization for quantum graphs with a random potential

By Carsten Schubert


We prove spectral localization for infinite metric graphs with a self-adjoint Laplace operator and a random potential. To do so we adapt the multiscale analysis (MSA) from the R^d-case to metric graphs. In the MSA a covering of the graph is needed which is obtained from a uniform polynomial growth of the graph. The geometric restrictions of the graph include a uniform bound on the edge lengths. As boundary conditions we allow all local settings which give a lower bounded self-adjoint operator with an associated quadratic form. The result is spectral localization (i.e. pure point spectrum) with polynomially decaying eigenfunctions in a small interval at the ground state energy.Comment: 56 page

Topics: Mathematics - Spectral Theory, Mathematical Physics, 82B44, 81Q10, 81Q35, 47B80
Year: 2012
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