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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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Rigidity of eigenvalues of generalized Wigner matrices
Consider N×N Hermitian or symmetric random matrices H with independent entries, where the distribution of the (i,j) matrix element is given by the probability measure νij with zero expectation andExpand
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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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The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics
Starting with a “relativistic” Schrödinger Hamiltonian for neutral gravitating particles, we prove that as the particle numberN→∞ and the gravitation constantG→0 we obtain the well knownExpand
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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 matrixExpand
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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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Bulk universality for generalized Wigner matrices
Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure νij with a subexponential decay. Let $${\sigma_{ij}^2}$$Expand
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Derivation of the nonlinear Schr\"odinger equation from a many body Coulomb system
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 theExpand
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The local semicircle law for a general class of random matrices
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a localExpand
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