Spectral minimal partitions of unbounded metric graphs

TL;DR

This paper studies spectral minimal partitions of unbounded metric graphs with Schrödinger operators, establishing conditions for their existence based on the essential spectrum, and illustrating these with various examples.

Contribution

It links spectral minimal partitions to the essential spectrum of Schrödinger operators on unbounded graphs and provides criteria for their existence or non-existence.

Findings

Minimal partitions are bounded above by the infimum of the essential spectrum.

Existence of minimal partitions occurs when the infimal energy is below the essential spectrum.

Examples demonstrate both existence and non-existence scenarios.

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.