# An optimization problem for finite point interaction families

## TL;DR

This paper investigates how to optimally arrange point interactions on manifolds like circles and spheres to maximize the ground state energy of associated Schrödinger operators, revealing optimal configurations for specific cases.

## Contribution

It provides a detailed analysis of optimal point configurations on symmetric manifolds for maximizing ground state energy, including solutions for spheres and insights into one-dimensional cases.

## Key findings

- Equal spacing maximizes eigenvalue on a circle.
- For spheres, optimal configurations relate to solutions of a modified Thomson problem.
- Equidistant distributions maximize eigenvalue for attractive interactions on an interval.

## Abstract

We consider the spectral problem for a family of $N$ point interactions of the same strength confined to a manifold with a rotational symmetry, a circle or a sphere, and ask for configurations that optimize the ground state energy of the corresponding singular Schr\"odinger operator. In case of the circle the principal eigenvalue is sharply maximized if the point interactions are distributed at equal distances. The analogous question for the sphere is much harder and reduces to a modification of Thomson problem; we have been able to indicate the unique maximizer configurations for $N=2,\,3,\,4,\,6,\,12$. We also discuss the optimization for one-dimensional point interactions on an interval with periodic boundary conditions. We show that the equidistant distributions give rise to maximum ground state eigenvalue if the interactions are attractive, in the repulsive case we get the same result for weak and strong coupling and we conjecture that it is valid generally.