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 Γ 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 Schrödinger operator in L(R), d = 2, 3, with N identical point interaction placed at a loop in the described way, and ask about the configuration which maximizes the ground state energy. We reduce both of them to geometric inequalities which involve chords of Γ; it will be shown that a sharp local extremum is in both cases reached by Γ in the form of a regular (planar) polygon and that such a Γ solves the two problems also globally.