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

Abstract

We consider a class of Hamiltonians in $L^2({\mathbb{R}}^2)$ 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 $\Gamma$.

LaTeX, 16 pages

References in corpus (4)

• Geometrically induced spectrum in curved leaky wires
• Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop
• Leaky quantum graphs: approximations by point interaction Hamiltonians
• An isoperimetric problem for point interactions

Cited by in corpus (5)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Inequalities for means of chords, with application to isoperimetric problems
• Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
• Spectral theory for Schrödinger operators with $\delta$-interactions supported on curves in ${\mathbb{R}}^3$
• On the critical exponent in an isoperimetric inequality for chords