05-34 Pavel Exner

An isoperimetric problem for leaky loops and related mean-chord inequalities (42K, LaTeX) Jan 26, 05

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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 $\Gamma$.

Files: 05-34.tex