# Spectral optimization for strongly singular Schr\"odinger operators with a star-shaped interaction

## TL;DR

This paper investigates the spectral properties of three-dimensional Schrödinger operators with star-shaped interactions, revealing that the principal eigenvalue is maximized in specific geometric configurations related to the Thomson problem.

## Contribution

It introduces a spectral optimization result showing the maximum principal eigenvalue occurs in five known star configurations, linking spectral theory with geometric arrangements.

## Key findings

- Principal eigenvalue maximized in five specific star configurations.
- Discrete spectrum exists when star arms are sufficiently long.
- Spectral properties depend on geometric arrangement of the star arms.

## Abstract

We discuss the spectral properties of singular Schr\"odinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maxi\-mized when the arms are arranged in one of the known five sharp configurations known as solutions of the closely related Thomson problem.