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Texas A&M University

Mathematical Physics and Harmonic Analysis Seminar

Date: November 8, 2019

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Gregory Berkolaiko, Texas A&M University


Title: Quantum graphs with a shrinking subgraph and exotic eigenvalues

Abstract: We address the question of convergence of Schroedinger 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. The failure is due to presence of what we call "exotic eigenvalues": eigenvalues whose eigenfunctions increasingly localize on the edges that are shrinking to a point.

We establish a sufficient condition for convergence which 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. In some important special cases this condition is also shown to be necessary. Moreover, when the condition fails, it provides quantitative information on exotic eigenvalues.

Before formulating the main results we will review the setting of Schrodinger operators on metric graphs and the characterization of possible self-adjoint conditions, followed by numerous examples where the limiting operator is not obvious or where the convergence fails outright. The talk is based on a joint work with Yuri Latushkin and Selim Sukhtaiev, arXiv:1806.00561 (Adv. Math. 2019) and on work in progress with Yves Colin de Verdiere.