Dirac particles on periodic quantum graphs

Dirac operators on manifolds with periodic ends

This paper studies Dirac operators on end-periodic spin manifolds of dimension at least 4. We provide a necessary and sufficient condition for such an operator to be Fredholm for a generic end-periodic metric. We make use of end-periodic Dirac operators to give an analytical interpretation of an invariant of non-orientable smooth 4-manifolds due to Cappell and Shaneson. From this interpretation...

Dirac Particles in Twisted Tubes

We consider the dynamics of a relativistic Dirac particle constrained to move in the interior of a twisted tube by confining boundary conditions, in the approximation that the curvature of the tube is small and slowly varying. In contrast with the nonrelativistic theory, which predicts that a particle's spin does not change as the particle propagates along the tube, we find that the angular mom...

Random Walks on Periodic Graphs

This paper concerns random walks on periodic graphs embedded in the d-dimensional Euclidian space Rd and obtains asymptotic expansions of the Green functions of them up to the second order term, which, expressed fairly explicitly, are easily computable for many examples. The result is used to derive an asymptotic form of the hitting distribution of a hyperplane of codimension one, which involve...

Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

Periodic orbit theory and spectral statistics for scaling quantum graphs

The explicit solution to the spectral problem of quantum graphs found recently in [20], is used to produce the exact periodic orbit theory description for the probability distributions of spectral statistics, including the distribution for the nearest neighbor separations, s n = k n − k n−1 , and the distribution of the spectral oscillations around the average, δk n = k n − ¯ k n .