Resonances in transport through leaky graphs
(by Pavel Exner)
In this lecture we will discuss generalized
Schr\"odinger operators given formally as -Δ -α δ(
x-Γ) in L2(R2), where Γ is a set of
measure zero. They can be regarded as models of nanostructures
taking the tunneling effect into account. Using an approximation
theorem indications are found that for a suitable geometry of
Γ such Hamiltonians exhibit resonances in the negative part
of the continuous spectrum. Furthermore, a solvable model will be
presented in which Γ consists of a line and a finite family
of points; using a resolvent formula we can reformulate it as a
Friedrichs-type problem and to find asymptotic properties of the
resonance poles. We will also discuss Zeno dynamics in this case.