The Experts below are selected from a list of 7143 Experts worldwide ranked by ideXlab platform
Zeev Rudnick - One of the best experts on this subject based on the ideXlab platform.
-
small scale equidistribution of eigenfunctions on the Torus
Communications in Mathematical Physics, 2017Co-Authors: Stephen Lester, Zeev RudnickAbstract:We study the small scale distribution of the L2 mass of eigenfunctions of the Laplacian on the Flat Torus \({\mathbb{T}^{d}}\). Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose L2 mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions d = 3, 4 we can restrict to individual eigenspaces and show small scale equidistribution in that context. We also study irregularities of quantum equidistribution: We construct eigenfunctions whose L2 mass does not equidistribute at all scales above the Planck scale. Additionally, in dimension d = 4 we show the existence of eigenfunctions for which the proportion of L2 mass in small balls blows up at certain scales.
-
nodal intersections for random eigenfunctions on the Torus
American Journal of Mathematics, 2016Co-Authors: Zeev Rudnick, Igor WigmanAbstract:We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional Flat Torus (``arithmetic random waves'') with a fixed smooth reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. In particular, this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability.
-
nodal intersections for random eigenfunctions on the Torus
arXiv: Mathematical Physics, 2014Co-Authors: Zeev Rudnick, Igor WigmanAbstract:We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional Flat Torus ("arithmetic random waves") with a fixed real-analytic reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behaviour of the nodal intersections variance for analytic curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. In particular, this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability.
-
restriction of toral eigenfunctions to hypersurfaces and nodal sets
Geometric and Functional Analysis, 2012Co-Authors: Jean Bourgain, Zeev RudnickAbstract:We give uniform upper and lower bounds for the L2 norm of the restriction of eigenfunctions of the Laplacian on the three-dimensional standard Flat Torus to surfaces with non-vanishing curvature. We also present several related results concerning the nodal sets of eigenfunctions.
-
statistics of wave functions for a point scatterer on the Torus
Communications in Mathematical Physics, 2012Co-Authors: Zeev Rudnick, Henrik UeberscharAbstract:Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a Flat Torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.
Igor Wigman - One of the best experts on this subject based on the ideXlab platform.
-
planck scale mass equidistribution of toral laplace eigenfunctions
Communications in Mathematical Physics, 2017Co-Authors: Andrew Granville, Igor WigmanAbstract:We study the small scale distribution of the L 2-mass of eigenfunctions of the Laplacian on the two-dimensional Flat Torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300, 2017) showed the existence of a density one subsequence whose L 2-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the L 2-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.
-
planck scale mass equidistribution of toral laplace eigenfunctions
arXiv: Number Theory, 2016Co-Authors: Andrew Granville, Igor WigmanAbstract:We study the small scale distribution of the $L^2$-mass of eigenfunctions of the Laplacian on the the two-dimensional Flat Torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick showed the existence of a density one subsequence whose $L^2$-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the $L^2$-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.
-
nodal intersections for random eigenfunctions on the Torus
American Journal of Mathematics, 2016Co-Authors: Zeev Rudnick, Igor WigmanAbstract:We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional Flat Torus (``arithmetic random waves'') with a fixed smooth reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. In particular, this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability.
-
nodal intersections for random eigenfunctions on the Torus
arXiv: Mathematical Physics, 2014Co-Authors: Zeev Rudnick, Igor WigmanAbstract:We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional Flat Torus ("arithmetic random waves") with a fixed real-analytic reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behaviour of the nodal intersections variance for analytic curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. In particular, this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability.
Andrew Granville - One of the best experts on this subject based on the ideXlab platform.
-
planck scale mass equidistribution of toral laplace eigenfunctions
Communications in Mathematical Physics, 2017Co-Authors: Andrew Granville, Igor WigmanAbstract:We study the small scale distribution of the L 2-mass of eigenfunctions of the Laplacian on the two-dimensional Flat Torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300, 2017) showed the existence of a density one subsequence whose L 2-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the L 2-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.
-
planck scale mass equidistribution of toral laplace eigenfunctions
arXiv: Number Theory, 2016Co-Authors: Andrew Granville, Igor WigmanAbstract:We study the small scale distribution of the $L^2$-mass of eigenfunctions of the Laplacian on the the two-dimensional Flat Torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick showed the existence of a density one subsequence whose $L^2$-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the $L^2$-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.
Jean-françois Cornet - One of the best experts on this subject based on the ideXlab platform.
-
A novel experimental bench dedicated to the accurate radiative analysis of photoreactors: The case study of CdS catalyzed hydrogen production from sacrificial donors
Chemical Engineering and Processing: Process Intensification, 2015Co-Authors: G. Dahi, A. Eskandari, J. Dauchet, F. Gros, M. Roudet, Jean-françois CornetAbstract:This article is dedicated to the presentation of a novel experimental bench designed to study the photoproduction of H2. It is composed of three main parts: a light source, a fully equipped Flat Torus reactor and the related analytical system. The reactor hydrodynamic behaviour has been carefully examined and it can be considered as perfectly mixed. The photon flux density is accurately known thanks to reconciled quantum sensor and actinometry experiments. The incident photon direction is perpendicular to the reactor windows; in such a configuration the radiative transfer description may be properly approximated as a one dimensional problem in Cartesian geometry. Based on accurate pressure measurement in the gas tight photoreactor, the production rates of H 2 (using CdS particles in association with sulphide and sulfite ions as hole scavengers) are easily and trustingly obtained. First estimations of apparent quantum yield have proven to be dependent on mean volumetric rate of radiant light energy absorbed hence demonstrating the need for the use of a radiative transfer approach to understand the observed phenomena and for the proper formulation of the thermo-kinetic coupling.
Naser T. Sardari - One of the best experts on this subject based on the ideXlab platform.
-
Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori
Communications in Mathematical Physics, 2018Co-Authors: Naser T. SardariAbstract:Let Q ( X ) be any integral primitive positive definite quadratic form in k variables, where $${k\geq4}$$ k ≥ 4 , and discriminant D . For any integer n , we give an upper bound on the number of integral solutions of Q ( X ) = n in terms of n , k , and D . As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the Flat Torus $${\mathbb{T}^d}$$ T d for $${d\geq 5}$$ d ≥ 5 . This conjecture is motivated by the work of Berry [ 2 , 3 ] on the semiclassical eigenfunction hypothesis.
-
quadratic forms and semiclassical eigenfunction hypothesis for Flat tori
arXiv: Number Theory, 2016Co-Authors: Naser T. SardariAbstract:Let $Q(X)$ be any integral primitive positive definite quadratic form with discriminant $D$ and in $k$ variables where $k\geq4$. We give an upper bound on the number of integral solutions of $Q(X)=n$ for any integer $n$ in terms of $n$, $k$ and $D$. As a corollary, we give a definite answer to a conjecture of Rudnick and Lester on the small scale equidistribution of orthonormal basis of eigenfunctions restricted to an individual eigenspace on the Flat Torus $\mathbb{T}^d$ for $d\geq 5$. Another application of our main theorem gives a sharp upper bound on $A_{d}(n,t)$, the number of representation of the positive definite quadratic form $Q(x,y)=nx^2+2txy+ny^2$ as a sum of squares of $d\geq 5$ linear forms where $n- n^{\frac{1}{(d-1)}-o(1)}< t < n$. This upper bound allows us to study the local statistics of integral points on sphere.