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

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

2007-11-27

arxiv.org/abs/0711.4247v1

Physics

Mathematical Physics

LaTeX, 15 pages

Scientific paper

10.1088/1751-8113/41/6/065305

We investigate relations between spectral properties of a single-centre point-interaction 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, $(-\Delta_{\Omega}^{D}+z) ^{-1}$. 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 $\Omega$, in particular, we establish some conditions showing how to place the interaction to optimize the principal eigenvalue.

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