Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems
- L. Erdős, B. Schlein, H. Yau
- Mathematics, Physics
- 2 August 2005
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…
Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate
- L. Erdős, B. Schlein, H. Yau
- Mathematics
- 2 October 2004
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 HN…
Derivation of the Gross‐Pitaevskii hierarchy for the dynamics of Bose‐Einstein condensate
- L. Erdős, B. Schlein, H. Yau
- Mathematics
- 2 October 2004
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 the…
Isotropic local laws for sample covariance and generalized Wigner matrices
- Alex Bloemendal, L. Erdős, A. Knowles, H. Yau, J. Yin
- Mathematics
- 27 August 2013
We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that…
The local semicircle law for a general class of random matrices
- L. Erdős, A. Knowles, H. Yau, J. Yin
- Mathematics
- 1 December 2012
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 local…
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 $${\sigma_{ij}^2}$$…
Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential
- L. Erdős, B. Schlein, H. Yau
- Mathematics
- 26 February 2008
Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let…
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…
Wegner estimate and level repulsion for Wigner random matrices
- L. Erdős, B. Schlein, H. Yau
- Mathematics
- 16 November 2008
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 consecutive…
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