Quantum Chaos and Anarchy

Sean P Gomes

May 3, 2012

Quantum chaos can be described as the study of the extent of the correlation between chaoticdynamics in a classical Hamiltonian system and chaotic dynamics in its quantum analogue.

An interesting example comes from ergodic billiards. Consider the geodesic flow on a compactRiemannian manifold M . The classical notion of chaos may be interpreted as the statement thatthis flow is ergodic with respect to the Liouville measure. For compact hyperbolic manifolds, orAnosov manifolds, billiard flow is necessarily ergodic.

The quantised version of billiards is a wave function onM that evolves according to Schrodinger’sequation and some appropriate boundary conditions. Such wave functions can be expanded as se-ries of eigenfunctions of the Laplace–Beltrami operator.

These eigenfunctions ψk give rise to a sequence of measures on M after L2-normalisation, hereit is understood that we order the eigenfunctions by increasing eigenvalue size. The quantumanalogue to the classical notion of ergodicity is the existence of a density 1 subsequence of thesemeasures which converges to the uniform measure in the weak-∗ sense .

A key result is the quantum ergodicity theorem which asserts that a compact Riemannianmanifold with an ergodic geodesic flow is quantum ergodic, in the sense outlined above.

Rudnick and Sarnak have further conjectured that the full sequence of measures in the aboveparagraph converges to the uniform measure when M is an Anosov manifold. This strongercondition is known as quantum unique ergodicity (QUE). Lindenstrauss proved the QUE conjecturefor some arithmetic surfaces in his prizewinning work.

Whilst the QUE conjecture is still open, progress has been made via bounds on the Kolmogorov–Sinai entropy of semiclassical measures. Such bounds give us information about the localisationproperties of eigenfunctions [1].

In contrast to the QUE conjecture made for Anosov manifolds, the planar stadium billiardprovides an example of a billiard that is ergodic but not QUE [2]. For this billiard an exceptionalsequence of eigenfunctions can be found which concentrate in the central rectangle.

For my research, I propose to investigate the localisation of eigenfunctions in the high energylimit, both over manifolds of Anosov type and over partially rectangular domains.

In particular, one question I would like to investigate is:

• “How chaotic does the billiard flow on a surface needs to be in order for the conjecturedQUE to hold? If we loosen our Anosov constraints on M to allow some ‘flat’ parts such as acylinder, is it possible to find eigenfunctions that concentrate purely in the cylinder?”.

References

[1] Nalini Anantharaman and Stephane Nonnenmacher. Chaotic vibrations and strong scars. 2010.

[2] Andrew Hassell. Ergodic billiards that are not quantum unique ergodic. Annals of Mathematics,2010.

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