Limits of Quantum Graph Operators With Shrinking Edges

Abstract

We address the question of convergence of Schrödinger 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.

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• NLS Bifurcations on the bowtie combinatorial graph and the dumbbell metric graph
• Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
• Magnetic ring chains with vertex coupling of a preferred orientation
• Exotic eigenvalues of shrinking metric graphs
• Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge
• Analyticity of resolvents of elliptic operators on quantum graphs with small edges
• Dirac particles on periodic quantum graphs
• Normalized NLS ground states on a double plane hybrid
• Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core