Pierre Duclos, Pavel Exner, Ondrej Turek
On the spectrum of a bent chain graph
(371K, pdf)
ABSTRACT. We study Schr\"odinger operators on an infinite quantum graph of a
chain form which consists of identical rings connected at the
touching points by $\delta$-couplings with a parameter
$\alpha\in\R$. If the graph is ``straight'', i.e. periodic with
respect to ring shifts, its Hamiltonian has a band spectrum with
all the gaps open whenever $\alpha\ne 0$. We consider a
``bending'' deformation of the chain consisting of changing one
position at a single ring and show that it gives rise to
eigenvalues in the open spectral gaps. We analyze dependence of
these eigenvalues on the coupling $\alpha$ and the ``bending
angle'' as well as resonances of the system coming from the
bending. We also discuss the behaviour of the eigenvalues and
resonances at the edges of the spectral bands.