207 research outputs found
Three-point correlations for quantum star graphs
We compute the three point correlation function for the eigenvalues of the
Laplacian on quantum star graphs in the limit where the number of edges tends
to infinity. This extends a work by Berkolaiko and Keating, where they get the
2-point correlation function and show that it follows neither Poisson, nor
random matrix statistics. It makes use of the trace formula and combinatorial
analysis.Comment: 10 pages, 2 figure
Form factor for large quantum graphs: evaluating orbits with time-reversal
It has been shown that for a certain special type of quantum graphs the
random-matrix form factor can be recovered to at least third order in the
scaled time \tau using periodic-orbit theory. Two types of contributing pairs
of orbits were identified, those which require time-reversal symmetry and those
which do not. We present a new technique of dealing with contribution from the
former type of orbits.
The technique allows us to derive the third order term of the expansion for
general graphs. Although the derivation is rather technical, the advantages of
the technique are obvious: it makes the derivation tractable, it identifies
explicitly the orbit configurations which give the correct contribution, it is
more algorithmical and more system-independent, making possible future
applications of the technique to systems other than quantum graphs.Comment: 25 pages, 14 figures, accepted to Waves in Random Media (special
issue on Quantum Graphs and their Applications). Fixed typos, removed an
overly restrictive condition (appendix), shortened introductory section
Diagonal approximation of the form factor of the unitary group
The form factor of the unitary group U(N) endowed with the Haar measure
characterizes the correlations within the spectrum of a typical unitary matrix.
It can be decomposed into a sum over pairs of ``periodic orbits'', where by
periodic orbit we understand any sequence of matrix indices. From here the
diagonal approximation can be defined in the usual fashion as a sum only over
pairs of identical orbits. We prove that as we take the dimension to
infinity, the diagonal approximation becomes ``exact'', that is converges to
the full form factor.Comment: 9 page
Quantum ergodicity for graphs related to interval maps
We prove quantum ergodicity for a family of graphs that are obtained from
ergodic one-dimensional maps of an interval using a procedure introduced by
Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take
the L^2 functions on the interval. The proof is based on the periodic orbit
expansion of a majorant of the quantum variance. Specifically, given a
one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an
increasingly refined sequence of partitions of the interval. To this sequence
we associate a sequence of graphs, whose directed edges correspond to elements
of the partitions and on which the classical dynamics approximates the
Perron-Frobenius operator corresponding to the map. We show that, except
possibly for subsequences of density 0, the eigenstates of the quantum graphs
equidistribute in the limit of large graphs. For a smaller class of observables
we also show that the Egorov property, a correspondence between classical and
quantum evolution in the semiclassical limit, holds for the quantum graphs in
question.Comment: 20 pages, 1 figur
No quantum ergodicity for star graphs
We investigate statistical properties of the eigenfunctions of the
Schrodinger operator on families of star graphs with incommensurate bond
lengths. We show that these eigenfunctions are not quantum ergodic in the limit
as the number of bonds tends to infinity by finding an observable for which the
quantum matrix elements do not converge to the classical average. We further
show that for a given fixed graph there are subsequences of eigenfunctions
which localise on pairs of bonds. We describe how to construct such
subsequences explicitly. These constructions are analogous to scars on short
unstable periodic orbits.Comment: 26 pages, 5 figure
Intermediate wave-function statistics
We calculate statistical properties of the eigenfunctions of two quantum
systems that exhibit intermediate spectral statistics: star graphs and Seba
billiards. First, we show that these eigenfunctions are not quantum ergodic,
and calculate the corresponding limit distribution. Second, we find that they
can be strongly scarred by short periodic orbits, and construct sequences of
states which have such a limit. Our results are illustrated by numerical
computations.Comment: 4 pages, 3 figures. Final versio
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