Effective Schroedinger dynamics on $ \epsilon $-thin Dirichlet waveguides via Quantum Graphs I: star-shaped graphs

TL;DR

This paper develops a method to approximate quantum dynamics in thin star-shaped waveguides using quantum graphs, focusing on boundary conditions that capture low-energy behavior.

Contribution

It introduces a specific set of boundary conditions at the vertex to accurately model low-energy quantum evolution in thin waveguides via quantum graphs.

Findings

Boundary conditions derived for low-energy quantum dynamics.

Effective quantum graph models for thin star-shaped waveguides.

Enhanced understanding of waveguide-to-graph approximations.

Abstract

We describe the boundary conditions at the vertex that one must choose to obtain a dynamical system that best describes the low-energy part of the evolution of a quantum system confined to a very small neighbourhood of a star-shaped metric graph.