An isoperimetric problem for point interactions

TL;DR

This paper investigates how the arrangement of point interactions in a Hamiltonian affects the ground state energy, showing that regular polygons locally maximize this energy and linking the problem to a geometric question.

Contribution

It demonstrates that for Hamiltonians with point interactions at polygon vertices, the regular polygon configuration locally maximizes the ground state energy, connecting spectral optimization with geometric properties.

Findings

Regular polygons locally maximize ground state energy.

The problem of global maximization is reduced to a geometric question.

The study links spectral properties with geometric configurations.

Abstract

We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all with the same coupling constant, placed at vertices of an equilateral polygon $\PP_N$. It is shown that the ground state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.