Recent progress on Quantum Unique Ergodicity (QUE)
by Peter Sarnak
October 3, 2008
Techniques for studying the fine behaviour of a highly excited state in the
quantization of classically chaotic hamiltonian, are limited. The basic general
tool is the theory of Fourier Integral Operators and related microlocal
analysis and it provides information
about averages of various quantities associated with these states.
Over the last 25 years, the development of fast numerical methods to compute
such states has exposed interesting phenomena and conjectures
(the corresponding study is called quantum chaos) but little of the finer
structure that is observed, can be proved.
Among the many interesting conjectures that have emerged is one
which asks about the distribution in the semiclassical limit of the familiar
probability measure ("Wigner measure") on phase space, that is associated with a
given quantum state. Any limit of these measures is called a quantum limit. The
behaviour in the semiclassical limit of almost all states for the quantization
of a classically ergodic hamiltonian mechanical system is one
of the few well understood phenomena in the subject. The quantum analogue of
ergodicity is the Quantum Ergodicity Theorem (due to
Shnirelman, Colin de Verdiere
and Zelditch) and it asserts that in the semiclassical limit, almost all these
states become equidistributed with respect to the corresponding classical
Lioville measure. The Quantum Unique Ergodicity Conjecture (Rudnick and Sarnak)
"QUE", asserts that for certain strongly chaotic hamiltonians (specifically
geodesic motion on the unit tangent space of a manifold of negative sectional
curvature) every state becomes equidistributed. This conjecture which was based
on insights from the arithmetic case (see below) was contrary to what was
suggested by some numerics, namely that some states may concentrate along
unstable periodic orbits (known as strong scars). QUE, if true, is in sharp
contrast to the behaviour of the classical trajectories for such chaotic
systems. It limits the extent to which the correspondence principle is valid
for chaotic systems.
In this generality the main breakthrough on the problem is due to
Nalini Anantharaman who has shown that strong scars cannot occur for
hamiltonians coming from the geodesic motion on a compact negatively curved
manifold. In fact she shows that any quantum limit must have positive entropy.
This is remarkable result whose proof combines ideas from microlocal analysis
with hyperbolic dynamics in a novel way.
Stadium billiards
Another recent advance by Andrew Hassell is concerned with
this equidistribution question for eigenstates on a "stadium".
The classical mechanics of this system, that is the billiard ball
motion in such a stadium, is not strongly chaotic (in the sense needed for the
QUE conjecture) and in particular there is the family of periodic orbits which
correspond to a billiard ball bouncing back and forth between the parallel
sides of the stadium. From the beginning numerical calculations suggested that
for this stadium there is a relatively sparse sequence of eigenstates that don't
become equidistributed and which correspond to these bouncing balls. A proof of
the existence of these bouncing ball modes is what Hassell has achieved.


A typical wave function.
Image from Douglas Stone's website
Bouncing Ball modes.
Image from Arnd Baecker's website

Arithmetic surfaces
The setting in which substantial progress has been made on
these questions of the behaviour of highly excited states in a chaotic
hamiltonian, is that corresponding to the geodesic motion a manifold of
negative curvature which is furthermore arithmetic (the study of
these problems in these cases is known as "arithmetic quantum chaos").
This of course is a very special type of manifold but it is one for which modern
techniques from number theory and automorphic forms can be used effectively to
investigate the eigenstates. The important feature defining these manifolds is
an infinite family of pseudosymmetries known as correspondences. These give
rise to geometric operators that commute with the quantization of the
Hamiltonian and in this setting one naturally assumes that the quantized states
are also eigenstates of these geometric "Hecke Operators". For these there is by
now a good understanding of the many of basic problems. In particular the
formula of
Thomas Watson, which explicitly connects periods of these eigenstates with the
theory of Lfunctions, is central to much of the progress. For example the QUE
conjecture follows for these states from something quite a bit weaker than the
Riemann Hypothesis for the corresponding Lfunctions (precisely "subconvexity"
suffices). Moreover the natural and interesting analogues of the QUE
conjecture, that is the equidistribution of mass of the norms of such
holomorphic hecke eigenforms (these are arithmetically perhaps the most
interesting modular forms) would also follow from the corresponding subconvexity
estimate.
The QUE problem for such Hecke states is now all but completely settled for
the case of arithmetic hyperbolic surfaces. First for compact such surfaces and
for eigenstates of the quantized hamiltonian (that is the Laplacian), QUE was
proven a few years ago by Elon Lindenstrauss. His proof introduces
novel ideas from ergodic theory, specifically measure rigidity on homogeneous
spaces (that is the classification of measures on such
spaces which are invariant under certian diagonal subgroups),
into the
analysis of quantum limits. While this approach has been
extended to handle the QUE conjecture for some higher dimensional
arithmetic manifolds (by Silberman and Venkatesh) it has not succeeded
in dealing with the analogue QUE problem for holomorphic forms
nor for noncompact arithmetic surfaces.
Many of the missing cases have now been resolved by Holowinsky and
Soundararajan. They have established the QUE conjecture for holomorphic
forms of increasing weight (this is the semiclassical limit in this setting) for
holomorphic Hecke forms on noncompact arithmetic surfaces. It is known in general for
compact complex manifolds [NonnenmacherVoros, ShiffmanZelditch]
and in the pertinent noncompact surface case[Rudnick], that the equidistribution
of mass of the norm of holomorphic sections of large
powers of a positive line bundle implies the equidistribution of its
zeros. Thus the zeros of a holomorphic Hecke eigenform on a noncompact
arithmetic surface become equidistributed with respect to the hyperbolic
area, as the weight goes to infinity. The last is a fundamental equidistribution
property in the arithmetic theory of modular forms and is of independent
interest.
The result of Holowinsky and Soundararajan comes from their separate attacks
on the holomorphic QUE problem. Each method achieves the desired result
up to a small number of exceptions. Remarkably their approaches are sufficiently
different so that the set of exceptions to both approaches, is empty! Their
proof relies heavily on the Ramanujan Conjectures which were established for
such holomorphic forms by Deligne in the 70's.
What remains unresolved as far as QUE in the case of arithmetic surfaces, is
the case of holomorphic forms on compact arithmetic surfaces and of eigenstates
of the Laplacian in the noncompact such surfaces.
A mathematical model setting for studying quantizations and semiclassical
limits is that of "quantizations "of symplectic maps. The special chaotic case
of a linear area preserving map of the two torus (ie an element of SL(2,Z)
whose trace has absolute value bigger than 2) goes by the name "a cat map". The
states of the corresponding quantization can be studied in depth. It turns out
that in this setting of cat maps, QUE can fail (FaureNonnenmacherDe Bievre).
Some take this as a warning about the truth of the original QUE
conjecture.
10/3/08
