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

Abstract

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.

LaTeX, 15 pages

References in corpus (2)

• Quantum mechanics of layers with a finite number of point perturbations
• Point Interaction Hamiltonians in Bounded Domains

Cited by in corpus (4)

• Self-adjoint extension schemes and modern applications to quantum Hamiltonians
• On the many Dirichlet Laplacians on a non-convex polygon and their approximations by point interactions
• Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions
• Resonances and resonance expansions for point interactions on the half-space