Antti Knowles

On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction

In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg- picture dynamics of “observables”, thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a “semi-classical” limit.

Mean-Field Dynamics: Singular Potentials and Rate of Convergence

We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N−1. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit N → ∞ the quantum N-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum N-body dynamics to the Hartree dynamics.

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

We consider the ensemble of adjacency matrices of Erdős–Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p≡p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN→∞ (with a speed at least logarithmic in N), the density of eigenvalues of the Erdős–Renyi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N−1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the l∞-norms of the l2-normalized eigenvectors are at most of order N−1/2 with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős–Renyi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN≫N2/3.

Eigenvector Distribution of Wigner Matrices

We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure νij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution νij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements Hxy, indexed by

On the principal components of sample covariance matrices

{x,y \in \Lambda \subset \mathbb{Z}^d}

The Altshuler–Shklovskii Formulas for Random Band Matrices I: the Unimodular Case

, are independent, uniformly distributed random variables if