Related papers: Arithmetic and equidistribution of measures on the…

Let $\Delta_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\Delta u + \lambda u =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate…

The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schr\"{o}dinger, Klein-Gordon and…

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-dimensional sphere ($d\ge 2$). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for…

In this paper we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish…

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $\mu$. Specifically, we characterize the measures $\mu$ for which the inequalities $$ \int…

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these…

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from…

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an…

The Wasserstein distance quantifies the distance between two probability measures on a metric space. We prove an analogue of the Berry-Esseen inequality for the Wasserstein distance on a finite area hyperbolic surface. This inequality…

We consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the…

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezin inequality…

Let $\phi$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of $\mathrm{PGL}_n$ over $\mathbb{Q}$ with Laplace eigenvalue $\lambda_{\phi}$. We establish explicit estimates for various…

This paper introduces an abstract spectral approach to prove effective equidistribution of expanding horospheres in hyperbolic manifolds. The method, which is motivated by the approach to counting developed by (Lax-Phillips 1982), produces…

We study the value distribution and extreme values of eigenfunctions for the ``quantized cat map''. This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of…

We show that for a positive proportion of Laplace eigenvalues $\lambda_j$ the associated Hecke-Maass $L$-functions $L(s,u_j)$ approximate with arbitrary precision any target function $f(s)$ on a closed disc with center in $3/4$ and radius…

We introduce quantum circuits in two and three spatial dimensions which are classically simulable, despite producing a high degree of operator entanglement. We provide a partial characterization of these "automaton" quantum circuits, and…

Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of…

For $2\leq p<4$, we study the $L^p$ norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact $2$-dimensional Riemannian manifolds. Burq, G\'erard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and…

We study the problem of mass distribution of Laplacian eigenfunctions in shrinking balls for the standard flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. By averaging over the centre of the ball we use Bourgain's de-randomisation to…

We consider an arbitrary selfadjoint operator on a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces…