Necklaces with interacting beads: isoperimetric problems

TL;DR

This paper explores two related isoperimetric problems involving point charges and quantum point interactions on closed loops, demonstrating that regular polygons optimize both energy minimization and maximization scenarios.

Contribution

It establishes that regular polygons uniquely solve both classical and quantum isoperimetric problems involving point interactions on loops.

Findings

Regular polygons are the extremizers for both problems.

Both problems reduce to geometric inequalities involving chords.

The solutions are globally optimal configurations.

Abstract

We discuss a pair of isoperimetric problems which at a glance seem to be unrelated. The first one is classical: one places $N$ identical point charges at a closed curve $\Gamma$ at the same arc-length distances and asks about the energy minimum, i.e. which shape does the loop take if left by itself. The second problem comes from quantum mechanics: we take a Schr\"odinger operator in $L^2(\mathbb{R}^d), d=2,3,$ with $N$ identical point interaction placed at a loop in the described way, and ask about the configuration which \emph{maximizes} the ground state energy. We reduce both of them to geometric inequalities which involve chords of $\Gamma$; it will be shown that a sharp local extremum is in both cases reached by $\Gamma$ in the form of a regular (planar) polygon and that such a $\Gamma$ solves the two problems also globally.