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

We investigate relations between spectral properties of a single-centre point-interaction Hamiltonian describing a particle confined to a bounded domain Ω subset of Rd, d=2,3, with a 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, (-ΔDΩ + 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