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

We analyze properties of semigroups generated by Schrödinger operators Δ−V or polyharmonic operators −(−Δ)m, on metric graphs both on Lp-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 Kreĭn-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.