If one attaches one or more infinite leads to a compact quantum
graph, most, but not necessarily all, of the eigenvalues turn into
resonances. This suggests that the leading asymptotics of the
resonance counting function (including any residual eigenvalues)
might obey the Weyl formula. Recent work with Pushnitski shows that
this is not always the case and provides a simple geometrical
condition for it to fail.  As well as outlining the proof of the
general theorem the lecture will describe a particular example in
some detail to explain how the transition from the Weyl to the
non-Weyl case occurs. It will also mention some more general contexts
in which the same phenomenon occurs, as discovered recently in work
with Exner and Lipovsky.

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<A HREF="Davies.pdf">
Here you will find a longish summary of the talk</A>.