MPHA Seminar, Blocker 627, 25 Jan 2008, 1.50pm

Speaker: Jon Harrison, Baylor
Title: Equi-transmitting scattering matrices for quantum graphs

We develop a new class of scattering matrices on graphs which have equal
transmission probabilities and no reflection amplitude.  This follows
previous generalizations of spectral problems on quantum graphs where
scattering matrices defined by an operator on the graph are replaced by
particular classes of unitary matrices.  Choosing matrices which do not
allow backscattering implies the set of periodic orbits summed over in
the trace formula is the same as that appearing in the definition of the
Ihara zeta function on the corresponding combinatorial graph.  We derive
properties of equi-transmitting graphs where the associated doubly
stochastic Markov matrices have a large spectral gap and hence the
spectral statistics are conjectured to converge rapidly to those of
random matrices.  Equi-transmitting matrices can be generated from
skew-Hadamard matrices or using Dirichlet characters.