## 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… Expand

## 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… Expand

## 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… Expand

## 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… Expand

## 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… Expand

## 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

## 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… Expand

## 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… Expand

## 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… Expand

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