Limits of Quantum Graph Operators With Shrinking Edges

TL;DR

This paper investigates the convergence behavior of Schrödinger operators on metric graphs with shrinking edges, identifying conditions under which the operators converge in a norm resolvent sense despite breakdowns of standard estimates.

Contribution

It provides a new sufficient condition for the convergence of quantum graph operators with shrinking edges, linking graph topology and vertex conditions.

Findings

Convergence depends on a balance between graph topology and vertex data.

Standard Sobolev estimates may fail as edges shrink, affecting convergence.

The convergence condition is independent of potential and edge length differences.

Abstract

We address the question of convergence of Schr\"odinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. We use a combination of functional-analytic bounds on the edges of the graph and Lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence. This condition encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges.