Point interaction polygons: An isoperimetric problem

Abstract

We will discuss a new type of an isoperimetric problem concerning a Hamiltonian with N point interactions in ℝd, d = 2, 3, all with the same coupling constant, placed at vertices of an equilateral polygon PN. We show that the ground state energy is locally maximized by a regular polygon and conjecture that the maximum is global; on the way we encounter an interesting geometric inequality. We will also mention some extensions of this problem. © Springer 2006.