Laplace and Schr\"odinger operators without eigenvalues on homogeneous amenable graphs

Rostislav Grigorchuk; Christophe Pittet

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arXiv · 2102.13542

Laplace and Schr\"odinger operators without eigenvalues on homogeneous amenable graphs

Rostislav Grigorchuk·Christophe Pittet

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Abstract

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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BibTeX RIS

Rostislav Grigorchuk, Christophe Pittet. 2021-02-26. Laplace and Schr\"odinger operators without eigenvalues on homogeneous amenable graphs. https://arxiv.org/abs/2102.13542

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