Necklaces with interacting beads: isoperimetric problems

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.