Spectral minimal partitions of unbounded metric graphs

Abstract

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 Schr\"odinger operator of the form $-\Delta + V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' 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 Schr\"odinger operator on the whole graph on the other. Namely, we show that for any $k\in\mathbb{N}$, 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 Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schr\"odinger operator-based partitions of unbounded domains in Euclidean space.