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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-linearExpand
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Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N(xi − xj)), where x = (x1, . . ., xN) denotes the positions of the particles. Let HNExpand
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Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate
Consider a system of N bosons on the three-dimensional unit torus interacting via a pair potential N2V(N(xi − xj)) where x = (x1, …, xN) denotes the positions of the particles. Suppose that theExpand
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Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N(xi − xj)), where x = (x1, . . . , xN) denotes the positions of the particles. Let HNExpand
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Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of theExpand
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Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. We study theExpand
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Wegner estimate and level repulsion for Wigner random matrices
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutiveExpand
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Universality of random matrices and local relaxation flow
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, weExpand
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Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons
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. RecallExpand
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Rigorous derivation of the Gross-Pitaevskii equation.
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 bosonicExpand
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