Yaiza Canzani

Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law

Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as {\lambda} tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most {\lambda}. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (\lambda, \lambda + 1] has a universal scaling limit as {\lambda} goes to infinity (depending only on the dimension of M). Our results also imply that if M has no conjugate points, then immersions of M into Euclidean space by an orthonormal basis of eigenfunctions with frequencies in (\lambda, \lambda + 1] are embeddings for all {\lambda} sufficiently large.

Nodal Sets of Schrödinger Eigenfunctions in Forbidden Regions

This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. In the case of real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region.

Fixed frequency eigenfunction immersions and supremum norms of random waves

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.

On the distribution of perturbations of propagated Schrödinger eigenfunctions

Let (M, g0) be a compact Riemmanian manifold of dimension n. Let P0(h) := −h∆g + V be the semiclassical Schrodinger operator for h ∈ (0, h0], and let E be a regular value of its principal symbol p0(x, ξ) = |ξ|2g0(x) + V (x). Write φh for an L -normalized eigenfunction of P (h), P0(h)φh = E(h)φh and E(h) ∈ [E − o(1), E + o(1)]. Consider a smooth family of perturbations gu of g0 with u in the ball B(e) ⊂ R of radius e > 0. For Pu(h) := −h∆gu + V and small |t|, we define the propagated perturbed eigenfunctions φ (u) h := e − i h φh. We study the distribution of the real part of the perturbed eigenfunctions regarded as random variables < “ φ (·) h (x) ” : B(e)→ R for x ∈M. In particular, when (M, g) is ergodic, we compute the h → 0 asymptotics of the variance Var h < “ φ (·) h (x) ”i and show that all odd moments vanish as h→ 0.

Averages of Eigenfunctions Over Hypersurfaces

AbstractLet (M, g) be a compact, smooth, Riemannian manifold and

CONFORMAL INVARIANTS FROM NODAL SETS. I. NEGATIVE EIGENVALUES AND CURVATURE PRESCRIPTION

{\{ \phi_h \}}

Local Integral Statistics for Monochromatic Random Waves

{ϕh} an L2-normalized sequence of Laplace eigenfunctions with defect measure