Quantum graphs as holonomic constraints

TL;DR

This paper studies how quantum dynamics on a graph can be approximated by a limiting process involving a shrinking potential well in R^2, showing convergence to free or Dirichlet Laplacian dynamics depending on the graph structure.

Contribution

It rigorously demonstrates the convergence of quantum dynamics on a graph to free or Dirichlet Laplacian dynamics as the potential well shrinks, providing a mathematical foundation for quantum graphs as holonomic constraints.

Findings

Convergence to free dynamics on edges for generic graphs

Convergence to Laplacian with Dirichlet conditions at vertices for simple graphs

Mathematical validation of quantum graphs as holonomic constraints

Abstract

We consider the dynamics on a quantum graph as the limit of the dynamics generated by a one-particle Hamiltonian in R^2 with a potential having a deep strict minimum on the graph, when the width of the well shrinks to zero. For a generic graph we prove convergence outside the vertices to the free dynamics on the edges. For a simple model of a graph with two edges and one vertex, we prove convergence of the dynamics to the one generated by the Laplacian with Dirichlet boundary conditions in the vertex.