167,376 research outputs found
One-dimensional quantum chaos: Explicitly solvable cases
We present quantum graphs with remarkably regular spectral characteristics.
We call them {\it regular quantum graphs}. Although regular quantum graphs are
strongly chaotic in the classical limit, their quantum spectra are explicitly
solvable in terms of periodic orbits. We present analytical solutions for the
spectrum of regular quantum graphs in the form of explicit and exact periodic
orbit expansions for each individual energy level.Comment: 9 pages and 4 figure
Quadratic maps with a periodic critical point of period 2
We provide a complete classification of possible graphs of rational
preperiodic points of endomorphisms of the projective line of degree 2 defined
over the rationals with a rational periodic critical point of period 2, under
the assumption that these maps have no periodic points of period at least 7. We
explain how this extends results of Poonen on quadratic polynomials. We show
that there are 13 possible graphs, and that such maps have at most 9 rational
preperiodic points. We provide data related to the analogous classification of
graphs of endomorphisms of degree 2 with a rational periodic critical point of
period 3 or 4.Comment: Updated theorem 2 to rule out the cases of quadratic maps with a
rational periodic critical point of period 2 and a rational periodic point of
period 5 or
A Jenkins-Serrin problem on the strip
We describe the family of minimal graphs on strips with boundary values
disposed alternately on edges of length one, and whose conjugate
graphs are contained in horizontal slabs of width one in . We can
obtain as limits of such graphs the helicoid, all the doubly periodic Scherk
minimal surfaces and the singly periodic Scherk minimal surface of angle
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
Trace formulae for d-regular graphs are derived and used to express the
spectral density in terms of the periodic walks on the graphs under
consideration. The trace formulae depend on a parameter w which can be tuned
continuously to assign different weights to different periodic orbit
contributions. At the special value w=1, the only periodic orbits which
contribute are the non back- scattering orbits, and the smooth part in the
trace formula coincides with the Kesten-McKay expression. As w deviates from
unity, non vanishing weights are assigned to the periodic walks with
back-scatter, and the smooth part is modified in a consistent way. The trace
formulae presented here are the tools to be used in the second paper in this
sequence, for showing the connection between the spectral properties of
d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure
Spectral statistics for unitary transfer matrices of binary graphs
Quantum graphs have recently been introduced as model systems to study the
spectral statistics of linear wave problems with chaotic classical limits. It
is proposed here to generalise this approach by considering arbitrary, directed
graphs with unitary transfer matrices. An exponentially increasing contribution
to the form factor is identified when performing a diagonal summation over
periodic orbit degeneracy classes. A special class of graphs, so-called binary
graphs, is studied in more detail. For these, the conditions for periodic orbit
pairs to be correlated (including correlations due to the unitarity of the
transfer matrix) can be given explicitly. Using combinatorial techniques it is
possible to perform the summation over correlated periodic orbit pair
contributions to the form factor for some low--dimensional cases. Gradual
convergence towards random matrix results is observed when increasing the
number of vertices of the binary graphs.Comment: 18 pages, 8 figure
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