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

TL;DR

This paper studies how the position of a point-interaction within a bounded domain affects the principal eigenvalue of the associated Hamiltonian, using explicit resolvent formulas and geometric analysis to optimize spectral properties.

Contribution

It provides explicit formulas for the resolvent of point-interaction Hamiltonians and characterizes how to optimally position the interaction to maximize the principal eigenvalue.

Findings

Explicit resolvent representation via Krein's formula.

Conditions for optimal placement of the point-interaction.

Analysis of ground-state energy behavior with respect to geometry.

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.