An isoperimetric problem for leaky loops and related mean-chord inequalities

TL;DR

This paper investigates a geometric inequality related to the maximization of the ground state of a Hamiltonian with attractive interactions supported on loops, showing the circle as a local maximizer and conjecturing a global maximizer.

Contribution

It introduces a new isoperimetric problem for leaky loops and connects it to mean-chord inequalities, proposing a conjecture about the global maximizer.

Findings

Ground state is locally maximized by a circle.

The problem relates to geometric inequalities of chord mean values.

Conjecture that the circle globally maximizes the ground state.

Abstract

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $\Gamma$ of a fixed length $L$, formally given by $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$. It is shown that the ground state of this operator is locally maximized by a circular $\Gamma$. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of $\Gamma$.