Random spherical harmonics and other
Riemannian random waves
June 15, 2011
Riemannian random waves
In this lecture, we:
Dene Riemannian random waves (of several kinds);
Show that their expected distribution of zeros is uniform;
same for random sequences o
CBMS Lecture
Lower bounds on hypersurface areas of nodal sets
Joint with C. Sogge
May 27, 2011
Nodal sets of eigenfunctions
Let (M , g ) be a compact C Riemannian manifold of dimesion n,
let be an L2 -normalized eigenfunction of the Laplacian,
= 2 ,
and
CBMS Lecture: Local Weyl laws
Steve Zelditch
Northwestern
Kentucky
June 20, 2011
Local Weyl law asymptotics
The classical Weyl law asymptotically counts the number of
eigenvalues less than ,
N () = #cfw_j : j =
|Bn |
Vol (M , g )n + O (n1 ).
(2 )n
(1)
He
Lp norms of eigenfunctions
Joint work with C. Sogge and J. Toth
CMBS Lecture
June, 2011
Lp norms
This lecture concerns the behavior as j of the Lp norms
|j |Lp
of L2 -normalized eigenfuntions. Let cfw_j (x ) be an orthonormal
basis of eigenfunctions and l
Quantum ergodicity
May 26, 2011
Asymptotics of matrix elements
How does Op (1E )j , j behave as j ? In the limit one
should get the classical probability that a particle of energy 2
j
lies in E Sg M . Ergodicity of the geodesic ow suggests that the
E|
lim
Weak * limit problem
CBMS Lecture
Expected value of an observable in an energy state
In quantum mechanics, the functional
j (A) = Aj , j
L2 (M )
is the expected value of the observable A in the energy state j
(energy = 2 ).
j
An observable is a bounded op
CBMS Lecture: Spectral rigidity of the ellipse
joint work with Hamid Hezari
CMBS Lecture
June, 2011
Isospectral rigidity of the ellipse
The purpose of this lecture is to prove that ellipses are spectrally
rigid among domains with their symmetries, i.e. do
Semi-classical oscillatory integral operators
References: Sogge, Fourier integrals in classical
analysis
Stein: Harmonic Analysis
Carleson-Sjolin and Hormander articles
CMBS Lecture
June, 2011
Semi-classical oscillatory integral operators
Let X be a manif
Eigenfunctions and eigenvalues on compact
surfaces of constant curvature
CMBS Lecture 2
June, 2011
Eigenfunctions
Let (M , g ) be a compact Riemannian manifold and let
1
g =
g
n
i ,j =1
xi
g ij g
xj
.
be its Laplace operator. (Note the -)
Let cfw_j be