LECTURE S. SOME APPLICATIONS OF SUBCONVEXITY 277ordered in increasing size: i.e. 6.cpj + Aj'Pj = 0 with 0 = >.. 0 ~ >.. 1 ~ >.. 2 ~
Considerations from theoretical physics (6. is the quantization of the Hamilton-
ian generating the geodesic flow) led to extensive investigations of the distribution
properties of 'Pj in the limit as >..j ----> +oo, and in particular, of the weak-* limits of
the sequence of probability measures
dμj = lcpj(x)l2dx
(here, dx is the normalized Riemannian volume)^2 ; such weak limits are called quan-
tum limits. When the geodesic flow is ergodic, an important result of Shnirelman,
Zelditch and Colin de Verdiere [Shn, Ze, C-V] shows that, at least for a full-density
subsequence {jk}k;;, 0 , dμJk weakly-* converges to dx. More precisely, one has for
any VE c=(X),(5 .12)this phenomenon is called Quantum Ergodicity. However, quantum ergodicity does
not exhibit an explicit subsequence having dx as its quantum limit, nor does it ex-
clude the possibility of having exceptional (zero density) subsequences dμJk having
a quantum limit different from dx. Such exceptional weak limits are called strong
scars and indeed have been observed numerically in some related chaotic dynami-
cal systems (such as billiards). In the special case of congruence hyperbolic surfaces
(i.e. quotients of H by congruence subgroups associated to quaternion algebras),
Rudnick and Sarnak [RS] have ruled out the existence of strong scars supported on
a finite union of points and closed geodesics (see also [BL] for a recent strengthen-
ing). This lead them to conjecture that in many cases dx is the only quantum limit
(Quantum Unique Ergodicity):QUE. Let X be a negatively curved compact manifold. Then dμj weakly converges to
dx as j----> +oo.
So far, the best evidence towards QUE comes from the case of arithmetic sur-
faces and arithmetic hyperbolic 3-folds. Indeed, one can then take advantage of the
extra symmetries provided by the (ergodic) action of the Hecke algebra. The study
of distribution properties of explicit sequences of primitive Hecke eigenforms is
sometimes called Arithmetic Quantum Chaos, and one of its most important conjec-
tures is to prove QUE for such Hecke eigenforms: the Arithmetic QUE conjecture.
Note that investigations of quantum limits are not limited to compact arithmetic
quotients (such as Shimura curves associated to congruence subgroups associated
to indefinite quaternion algebras) nor to Laplace eigenforms; for instance, Arith-
metic QUE for modular curve is as follows:
Arithmetic QUE. For any fixed q ~ 1, let f be a primitive weight zero Maass cusp
form (resp. holomorphic cusp form) -with nebentypus trivial or not-for the group
r 0 (q) with eigenvalue >.. 1 (resp. with weight k 1 J. Then as>..----> +oo (resp kt ----> +oo),
the measure
lf(z)l^2 dxdy lf(z)l^2 k dxdy
dμ 1 (z ) := (f, f) y (resp. dμ1(z ) := (f, j ) y J y)
(^2) d μ j is interpreted in quantum mechanics as the probability density for finding a particle in the state
"<p/' at the point x.