Mathematical Physics and Harmonic Analysis Seminar

Date: October 28, 2022

Time: 1:50PM - 2:50PM

Location: BLOC 306

Speaker: Matthias Hofmann, TAMU

Title: 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 $\lambda_{\text{ess}}$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other, which recalls a similar principle for the eigenvalues of the latter: for any $k\in\mathbb N$, the infimal energy among all admissible $k$-partitions is bounded from above by $\lambda_{\text{ess}}$, and if it is strictly below $\lambda_{\text{ess}}$, then a spectral minimal $k$-partition exists. We illustrate our results with several examples of existence and nonexistence 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. Joint project with James Kennedy and Andrea Serio.