Pavel Exner
Necklaces with interacting beads: isoperimetric problem
(35K, LaTeX)
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.