# Quantum graphs and dimensional crossover: the honeycomb

## TL;DR

This paper investigates the existence of ground states for the nonlinear Schrödinger equation on honeycomb-shaped metric graphs, revealing how the interplay of different dimensional scales causes threshold phenomena called dimensional crossover.

## Contribution

It extends previous results from square grid graphs to honeycomb graphs, demonstrating the impact of dimensional crossover on ground state existence.

## Key findings

- Identification of threshold phenomena in honeycomb graphs
- Extension of dimensional crossover results to new graph structures
- Insights into the interplay of 1D and 2D scales in quantum graphs

## Abstract

We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schr\"odinger Equation on doubly-periodic metric graphs. We extend the results known for the two--dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one--dimensional and two--dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover.