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Laplace and Schr\"odinger operators without eigenvalues on homogeneous amenable graphs
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A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schr\"odinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayley graph of (G,S') has pure point spectrum.
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Random Schr\"odinger operators and convolution on wreath products
Averaged spectral measures of a class of random Schrödinger operators coincide with Plancherel measures of convolution operators on wreath products, yielding new spectral criteria and Lifshitz tail estimates.
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