 07297 Pavel Exner and Andrea Mantile
 On the optimization of the principal eigenvalue for singlecentre pointinteraction operators in a bounded region
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Nov 27, 07

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Abstract. We investigate relations between spectral properties of a singlecentre
pointinteraction Hamiltonian describing a particle confined to a bounded domain $\Omega\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the geometry of $\Omega$. For this class of operators Krein's formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one, $\left( \Delta_{\Omega}^{D}+z\right) ^{1}$. We use a moving plane analysis to characterize the behaviour of the groundstate energy of the Hamiltonian with respect to the pointinteraction position and the shape of $\Omega$, in particular, we establish some conditions showing how to place the interaction to optimize the principal eigenvalue.
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