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...
01 -- 02 -- 03 -- 04 -- 05 -- 06 -- 07 -- 08 -- 09 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 -- 16

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Last modified: Jan 27 2004

Stefan Keppeler (quantum.ergodicityATmatfys.lth.se)