2005/08/31 by Pavel Exner, Exner, Pavel
Materials Science · Mathematics · Physics and Astronomy · #51P05 #78A30 #81V99 #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Mathematical Physics (math-ph) #Mathematics and Applications #Quantum Physics (quant-ph) #Quasicrystal Structures and Properties #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:51P05 #msc:78A30 #msc:81V99 #quant-ph
paper · pdf · doi:10.48550/arxiv.math-ph/0508061
AMSTeX, 9 pages
arxiv created 2005/08/31 · openalex publication_date 2005/08/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2(ℝd), 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.