Preface
These Proceedings are the outcome of an International Conference on Spectral
Geometry held at Dartmouth College on July 19-23, 2010. Over eighty gradu-
ate students, postdoctoral researchers, and senior researchers participated in the
conference, and many participants attended three minicourses on July 16-17 that
gave needed background on semiclassical analysis, spectral theory on hyperbolic
surfaces, and orbifold spectral geometry.
This volume contains these preparatory lectures together with a reprint of Peter
Sarnak’s article “Recent Progress on the Quantum Unique Ergodicity Conjecture”
which will provide valuable background for researchers interested in forefront de-
velopments in spectral geometry. Research contributions include recent work on
semiclassical measures, inverse spectral geometry, spectral properties of quantum
graphs, statistics of nodal lines of eigenfunctions, spectral asymptotics, and many
other developments of current interest in the field. We hope that researchers inter-
ested in spectral geometry and its interactions with number theory, physics, and
applied mathematics will find this collection a valuable reference.
One of the principal themes in the conference and in this volume is the behavior
of eigenfunctions. Peter Sarnak’s survey article describes the very exciting recent
progress on the Quantum Unique Ergodicity (QUE) conjecture of Zeev Rudnick
and Sarnak, which addresses the behavior of the highly excited states in the quan-
tization of ergodic Hamiltonian systems. Microlocal analysis allows one to associate
functions on a phase space T
∗M
to functions on the configuration space, M, for
example by the Wigner function construction. Weak limits of Wigner functions
of sequences {φj } of eigenfunctions of the Laplacian on a Riemannian manifold
measure whether the eigenfunctions tend to concentrate or become equidistributed
(tend to Liouville measure) in the semi-classical limit. The beautiful Quantum Er-
godicity Theorem says that if the classical flow is ergodic, there is a subsequence
of density one that is uniformly distributed. This theorem leaves open, however,
the possibility of “scarring”: Could there exist subsequences of density zero that
converge, say, to a measure concentrated along a periodic orbit of the Hamiltonian
system?
The survey article by Nalini Anantharaman and Fabricio Maci` a provides an in-
novative companion to Sarnak’s survey. The fundamental solution of the Schr¨odinger
operator without potential on a compact manifold, eitΔ, is very dispersive because
it propagates higher frequency waves faster than low frequency ones. It turns out
however that time averages of Wigner distributions of suitable functions of the
form
eitΔ(un),
where n is a large frequency parameter, have a semiclassical or
high-frequency limit. (Here the un’s are quite general; they need not be eigenfunc-
tions.) In their paper, Anantharaman and Maci` a review some of their results on
vii
Previous Page Next Page