Spectral minimal partitions of unbounded metric graphs

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrodinger operator of the form -A + V with suitable (electric) potential V, which is taken as a fixed, underlying function on the whole graph.We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum Sigma of the essential spectrum of the corresponding Schrodinger operator on the whole graph on the other. Namely, we show that for any k is an element of N, the infimal energy among all admissible k -partitions is bounded from above by Sigma and if it is strictly below Sigma then a spectral minimal k -partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials.The nature of the proofs, a key ingredient of which is a version of the characterization of the infimum of the essential spectrum known as Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrodinger operator-based partitions of unbounded domains in Euclidean space.