László Erdős

Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k.

Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation

We study the long time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to the solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum scattering cross section.

Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

We consider

Universality of Wigner random matrices: a survey of recent results

N\times N

Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order

Universality of local spectral statistics of random matrices

1/N

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

. We study the connection between eigenvalue statistics on microscopic energy scales

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

\eta\ll1

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order

THE KERNEL OF DIRAC OPERATORS ON

\eta \sim\log N/N