18 research outputs found
High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows
We relate high-energy limits of Laplace-type and Dirac-type operators to
frame flows on the corresponding manifolds, and show that the ergodicity of
frame flows implies quantum ergodicity in an appropriate sense for those
operators. Observables for the corresponding quantum systems are matrix-valued
pseudodifferential operators and therefore the system remains non-commutative
in the high-energy limit. We discuss to what extent the space of stationary
high-energy states behaves classically.Comment: 26 pages, latex2
Geodesics in the space of measure-preserving maps and plans
We study Brenier's variational models for incompressible Euler equations.
These models give rise to a relaxation of the Arnold distance in the space of
measure-preserving maps and, more generally, measure-preserving plans. We
analyze the properties of the relaxed distance, we show a close link between
the Lagrangian and the Eulerian model, and we derive necessary and sufficient
optimality conditions for minimizers. These conditions take into account a
modified Lagrangian induced by the pressure field. Moreover, adapting some
ideas of Shnirelman, we show that, even for non-deterministic final conditions,
generalized flows can be approximated in energy by flows associated to
measure-preserving maps
Level spacing statistics of classically integrable systems -Investigation along the line of the Berry-Robnik approach-
By extending the approach of Berry and Robnik, the limiting level spacing
distribution of a system consisting of infinitely many independent components
is investigated. The limiting level spacing distribution is characterized by a
single monotonically increasing function of the level spacing
. Three cases are distinguished: (i) Poissonian if ,
(ii) Poissonian for large , but possibly not for small if
, and (iii) sub-Poissonian if .
This implies that, even when energy-level distributions of individual
components are statistically independent, non-Poissonian level spacing
distributions are possible.Comment: 19 pages, 4 figures. Accepted for publication in Phys. Rev.
Local structure of the set of steady-state solutions to the 2D incompressible Euler equations
It is well known that the incompressible Euler equations can be formulated in
a very geometric language. The geometric structures provide very valuable
insights into the properties of the solutions. Analogies with the
finite-dimensional model of geodesics on a Lie group with left-invariant metric
can be very instructive, but it is often difficult to prove analogues of
finite-dimensional results in the infinite-dimensional setting of Euler's
equations. In this paper we establish a result in this direction in the simple
case of steady-state solutions in two dimensions, under some non-degeneracy
assumptions. In particular, we establish, in a non-degenerate situation, a
local one-to-one correspondence between steady-states and co-adjoint orbits.Comment: 81 page
Complex zeros of real ergodic eigenfunctions
We determine the limit distribution (as ) of complex
zeros for holomorphic continuations \phi_{\lambda}^{\C} to Grauert tubes of
real eigenfunctions of the Laplacian on a real analytic compact Riemannian
manifold with ergodic geodesic flow. If is an
ergodic sequence of eigenfunctions, we prove the weak limit formula
\frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial}
{\partial} |\xi|_g, where [Z_{\phi_{j_k}^{\C}}] is the current of
integration over the complex zeros and where is with respect
to the adapted complex structure of Lempert-Sz\"oke and Guillemin-Stenzel.Comment: Added some examples and references. Also added a new Corollary, and
corrected some typo
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
Approach to ergodicity in quantum wave functions
According to theorems of Shnirelman and followers, in the semiclassical limit
the quantum wavefunctions of classically ergodic systems tend to the
microcanonical density on the energy shell. We here develop a semiclassical
theory that relates the rate of approach to the decay of certain classical
fluctuations. For uniformly hyperbolic systems we find that the variance of the
quantum matrix elements is proportional to the variance of the integral of the
associated classical operator over trajectory segments of length , and
inversely proportional to , where is the Heisenberg
time, being the mean density of states. Since for these systems the
classical variance increases linearly with , the variance of the matrix
elements decays like . For non-hyperbolic systems, like Hamiltonians
with a mixed phase space and the stadium billiard, our results predict a slower
decay due to sticking in marginally unstable regions. Numerical computations
supporting these conclusions are presented for the bakers map and the hydrogen
atom in a magnetic field.Comment: 11 pages postscript and 4 figures in two files, tar-compressed and
uuencoded using uufiles, to appear in Phys Rev E. For related papers, see
http://www.icbm.uni-oldenburg.de/icbm/kosy/ag.htm