We obtain new quantitative estimates on Weyl Law remainders under dynamical
assumptions on the geodesic flow. On a smooth compact Riemannian manifold (M, g) of …

Here we give a survey with some detailed proofs on some basic analysis on Alexandrov
spaces. In particular, we sketched the proof of Sobolev and Poincaré inqualities on …

This work concerns $ L^ p $ norms of high energy Laplace eigenfunctions, $(-\Delta_g-
\lambda^ 2)\phi_\lambda= 0$, $\|\phi_\lambda\| _ {L^ 2}= 1$. In 1988, Sogge gave optimal …

Given a Riemannian manifold endowed with its Laplace-Beltrami operator, consider the
associated spectral projector on a thin interval. As an operator from L2 to Lp, what is its …

Let (M, g) be a compact Riemannian manifold of dimension n and P_1:=-h^ 2 Δ _g+ V (x)-
E_1 P 1:=-h 2 Δ g+ V (x)-E 1 so that dp_1 ≠ 0 dp 1≠ 0 on p_1= 0 p 1= 0. We assume that …

Abstract The classical Weyl Law says that if NM (λ) denotes the number of eigenvalues of
the Laplace operator on ad-dimensional compact manifold M without a boundary that are …

Let $(M, g) $ be a smooth, compact Riemannian manifold and ${\lbrace\phi\lambda\rbrace}
$ an $ L^ 2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g\phi_\lambda …

This work is concerned with operators of the type AD Qc acting in domains 0. 0; H/Â Rd RC:
The diffusion coefficient Qc> 0 depends on one coordinate y 2. 0; H/and is bounded but may …

This book aims to explain the concepts behind the geodesic beam method that we have
developed to study the behavior of high energy eigenfunctions. The idea for geodesic …

Let (M, g) be a smooth, compact, Riemannian manifold and {ϕ h} a sequence of L 2-
normalized Laplace eigenfunctions on M. For a smooth submanifold H⊂ M, we consider the …