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Now showing items 11-20 of 24
Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation
(Wiley-Blackwell, 2000)
We study the 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 a solution of a linear Boltzmann equation ...
Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
(Springer Nature, 2012)
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 that its bulk eigenvalues ...
Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams
(Springer Nature, 2007)
We consider random Schrödinger equations on {mathbb{R}d} for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ t the solution with initial data ψ0. The space and time ...
Isotropic local laws for sample covariance and generalized Wigner matrices
(Institute of Mathematical Statistics, 2014)
We consider sample covariance matrices of the form X ∗X, where X is an M × N matrix with independent random entries. We prove the isotropic local MarchenkoPastur law, i.e. we prove that the resolvent (X ∗X − z) −1 converges ...
Universality of local spectral statistics of random matrices
(American Mathematical Society (AMS), 2012-01-30)
The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix ...
On the Quantum Boltzmann Equation
(Springer Nature, 2004)
We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrödinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property ...
Fixed Energy Universality for Generalized Wigner Matrices
(Wiley-Blackwell, 2015)
We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require ...
Nonlinear Hartree Equation as the Mean Field Limit of Weakly Coupled Fermions
(Elsevier BV, 2004)
We consider a system of N weakly interacting fermions with a real analytic pair interaction. We prove that for a general class of initial data there exists a fixed time T such that the difference between the one particle ...
Quantum diffusion of the random Schrödinger evolution in the scaling limit
(International Press of Boston, 2008)
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 ψtψt the solution with initial data ψ0ψ0 . The space and time ...
Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons
(Springer Nature, 2005)
We consider the dynamics of N boson systems interacting through a pair potential N−1Va(xi−xj) where Va(x)=a−3V(x/a). We denote the solution to the N-particle Schrödinger equation by ΨN, t. Recall that the Gross-Pitaevskii ...