We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the… (More)

The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic… (More)

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so… (More)

We consider nonlinear Schrödinger equations in R. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions… (More)

We consider a nonlinear Schrödinger equation in R3 with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition.… (More)

We consider a general class of N × N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle… (More)

We consider random Schrödinger equations on Rd for d ≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The… (More)

We consider random Schrödinger equations on Zd for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time… (More)

We consider random Schrödinger equations on Rd for d ≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The… (More)

Let gc L ;; denote the grand canonical Gibbs measure of a lattice gas in a cube of size L with the chemical potential and a xed boundary condition. Let c L ;n be the corresponding canonical measure… (More)