Necklaces with interacting beads: isoperimetric problems
| dc.creator | Exner, Pavel | |
| dc.date | 2005-08-31 | |
| dc.date.accessioned | 2026-07-07T04:32:18Z | |
| dc.date.available | 2026-07-07T04:32:18Z | |
| dc.description | 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 $L^2(\mathbb{R}^d), d=2,3,$ 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 $Γ$; 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. | |
| dc.description | AMSTeX, 9 pages | |
| dc.identifier | https://arxiv.org/abs/math-ph/0508061 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0508061 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/58142 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Spectral Theory | |
| dc.subject | Quantum Physics | |
| dc.subject | 51P05, 81V99, 78A30 | |
| dc.title | Necklaces with interacting beads: isoperimetric problems | |
| dc.type | text |