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 Schrödinger 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 Schrödinger 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 Schrödinger operator-based partitions of unbounded domains in Euclidean space.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2022
- DOI:
- 10.48550/arXiv.2209.03658
- arXiv:
- arXiv:2209.03658
- Bibcode:
- 2022arXiv220903658H
- Keywords:
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- Mathematics - Spectral Theory;
- 34B45;
- 34L40;
- 35P15;
- 35R02;
- 81Q35
- E-Print:
- Revised, shortened version, a number of proofs and examples have been tightened. To appear in J. Spectr. Theory