Gregory Berkolaiko

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 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.

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 [math.SP].