Lund, 22nd - 24th January 2004
Quantum Ergodicity - Basic Ideas and Recent Developments
Introduction to quantum ergodicity and arithmetic quantum maps
Mini Course
Pär Kurlberg (Gothenburg)
The quantum mechanical counterpart to classical ergodicity is that
eigenfunctions of the quantized Hamiltonian (stationary states) should
be equidistributed in a certain sense. If a dynamical system is
classically ergodic, Schnirelman's theorem asserts that most
stationary states are equidistributed. However, subsequences of
exceptional eigenfunctions (scars) cannot be ruled out. On the other
hand, for some systems it seems reasonable to expect that no
exceptional eigenfunctions exist, so called Quantum Unique Ergodicity
(QUE). Other expected consequences of classical chaos is that the
value distribution of eigenfunctions, as well as the value
distribution of matrix coefficients of observables, should be
Gaussian.
We will discuss these topics in the context of cat maps, a simple
example of a chaotic dynamical system. Cat maps have strong
arithmetical properties, and as a result there can be very large
spectral degeneracies. In fact, when the spectral degeneracies are
huge, scarring actually occurs! On the other hand, spectral
degeneracies suggests that one should look for a family of commuting
operators. It turns out that such a family exists, and using this it
is possible to show that QUE holds for desymmetrized eigenfunctions.
Time permitting, we will make some comparisons with billiards on
modular surfaces, another system whose quantization has strong
arithmetical properties.
Watch the talk evolve on the black board...
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Workshop Home
Last modified: Jan 27 2004
Stefan Keppeler
(quantum.ergodicityATmatfys.lth.se)