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arXiv · 0711.4247

On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region

Pavel Exner·Andrea Mantile
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Abstract

We investigate relations between spectral properties of a single-centre point-interaction Hamiltonian describing a particle confined to a bounded domain $Ω\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the geometry of $Ω$. 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, $(-Δ_Ω^{D}+z) ^{-1}$. We use a moving plane analysis to characterize the behaviour of the ground-state energy of the Hamiltonian with respect to the point-interaction position and the shape of $Ω$, in particular, we establish some conditions showing how to place the interaction to optimize the principal eigenvalue.

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Pavel Exner, Andrea Mantile. 2007-11-27. On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region. https://doi.org/10.1088/1751-8113%2F41%2F6%2F065305

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