 0534 Pavel Exner
 An isoperimetric problem for leaky loops and related
meanchord inequalities
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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 $\Gamma$.
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