# Optimization of the lowest eigenvalue for leaky star graphs

## TL;DR

This paper proves that among star-shaped supports with fixed edge length and number, the symmetric star graph maximizes the lowest eigenvalue of a 2D Schrödinger operator with a delta interaction, using geometric and spectral analysis.

## Contribution

It establishes the optimality of symmetric star graphs for the lowest eigenvalue in a geometric spectral optimization problem.

## Key findings

- Symmetric star graphs maximize the lowest eigenvalue.
- The proof uses Birman-Schwinger principle and properties of Macdonald functions.
- A geometric inequality for polygons inscribed in a circle is employed.

## Abstract

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr\"odinger operator with an attractive $\delta$-interaction of a fixed strength, the support of which is a star graph with finitely many edges of an equal length $L \in (0,\infty]$. Under the constraint of fixed number of the edges and fixed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle.