Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra

Abstract

We analyze properties of semigroups generated by Schrödinger operators $-\hat{I}”+V$ or polyharmonic operators $-(-\hat{I}”{)}^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kre\uın-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

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