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Benjamin Schlein

Publications and source records attributed to Benjamin Schlein.

At least 91 records · Page 5Linked to original sources
arXiv2010

Ionization of Atoms by Intense Laser Pulses

The process of ionization of a hydrogen atom by a short infrared laser pulse is studied in the regime of very large pulse intensity, in the dipole approximation. Let $A$ denote the integral of the electric field of the pulse over time at the location of the atomic nucleus. It is shown that, in the limit where $|A| \to \infty$, the ionization probability approaches unity and the electron is ejected into a cone opening in the direction of $-A$ and of arbitrarily small opening angle. Asymptotics of various physical quantities in $|A|^{-1}$ is studied carefully. Our results are in qualitative agreement with experimental data reported in \cite{1,2}.

math-ph↗
arXiv2010

Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance $N^{-3/4+β}$ for some positive $β>0$. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.

math-ph↗
arXiv2009

Bulk Universality for Wigner Matrices

We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $ν(x)= e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $ν(x) \le C e^{- C |x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.

math-ph↗
arXiv2009

Derivation of effective evolution equations from many body quantum dynamics

We review some recent results concerning the derivation of effective evolution equations from first principle quantum dynamics. In particular, we discuss the derivation of the Hartree equation for mean field systems and the derivation of the Gross-Pitaevskii equation for the time evolution of Bose-Einstein condensates.

math-ph↗
arXiv2009

On Blowup for time-dependent generalized Hartree-Fock equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree-Fock-Bogoliubov theory.

math.AP↗
arXiv2009

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 the connection between eigenvalue statistics on microscopic energy scales $η\ll1$ and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order $η\sim\log N/N$. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales $η\gg N^{-2/3}$. We show that most eigenvectors are fully delocalized in the sense that their $\ell^p$-norms are comparable with $N^{{1}/{p}-{1}/{2}}$ for $p\ge2$, and we obtain the weaker bound $N^{{2}/{3}({1}/{p}-{1}/{2})}$ for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.

math-ph↗
arXiv2009

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 consecutive eigenvalues is of order $1/N$. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales $η\gg N^{-1}$. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result \cite{ESY2}. We then show a Wegner estimate, i.e. that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.

math-ph↗
arXiv2009

Derivation of the two dimensional nonlinear Schrodinger equation from many body quantum dynamics

We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.

math-ph↗
arXiv2009

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 $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $ψ_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $ψ_{N,0}$ satisfies the energy condition \[ < ψ_{N,0}, H_N ψ_{N,0} > \leq C N >. \] and that the one-particle density matrix converges to a projection as $N \to \infty$. Then, we prove that the $k$-particle density matrices of $ψ_{N,t}$ factorize in the limit $N \to \infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant proportional to the scattering length of the potential $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we develop a new approach that requires no restriction on the size of the potential.

math-ph↗
arXiv2009

The ground state energy of a low density Bose gas: a second order upper bound

Consider $N$ bosons in a finite box $Λ= [0,L]^3\subset \bR^3$ interacting via a two-body nonnegative soft potential $V= λ\tilde V$ with $\tilde V$ fixed and $λ>0$ small. We will take the limit $L, N \to \infty$ by keeping the density $\varrho= N/L^{3}$ fixed and small. We construct a variational state which gives an upper bound on the ground state energy per particle $\e$ $$ \e \le 4π\varrho a \Big [1+ \frac{128}{15\sqrtπ}(\varrho a^3)^{1/2}S_λ\Big ] + O(\varrho^2|\log\varrho|), \quad {as $\varrho\to 0$} $$ with a constant satisfying $$ 1\leq S_λ\leq 1+Cλ. $$ Here $a$ is the scattering length of $V$ and thus depends on $λ$. In comparison, the prediction by Lee-Yang \cite{LYang} and Lee-Huang-Yang \cite{LHY} asserts that $S_λ=1$ independent of $λ$.

math-ph↗
arXiv2008

Local semicircle law and complete delocalization for Wigner random matrices

We consider $N\times N$ Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales $η\gg N^{-1} (\log N)^8$. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the $\ell^\infty$-norm of the corresponding eigenvectors is of order $O(N^{-1/2})$, modulo logarithmic corrections. The upper bound $O(N^{-1/2})$ implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements.

math-ph↗
arXiv2008

Dynamical formation of correlations in a Bose-Einstein condensate

We consider the evolution of $N$ bosons interacting with a repulsive short range pair potential in three dimensions. The potential is scaled according to the Gross-Pitaevskii scaling, i.e. it is given by $N^2V(N(x_i-x_j))$. We monitor the behavior of the solution to the $N$-particle Schrödinger equation in a spatial window where two particles are close to each other. We prove that within this window a short scale interparticle structure emerges dynamically. The local correlation between the particles is given by the two-body zero energy scattering mode. This is the characteristic structure that was expected to form within a very short initial time layer and to persist for all later times, on the basis of the validity of the Gross-Pitaevskii equation for the evolution of the Bose-Einstein condensate. The zero energy scattering mode emerges after an initial time layer where all higher energy modes disperse out of the spatial window. We can prove the persistence of this structure up to sufficiently small times before three-particle correlations could develop.

math-ph↗
arXiv2008

Quantum Dynamics with Mean Field Interactions: a New Approach

We propose a new approach for the study of the time evolution of a factorized $N$-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the $N$-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.

math-ph↗
arXiv2008

Stellar Collapse in the Time Dependent Hartree-Fock Approximation

We prove blow-up in finite time for radially symmetric solutions to the pseudo-relativistic Hartree-Fock equation with negative energy. The non-linear Hartree-Fock equation is commonly used in the physics literature to describe the dynamics of white dwarfs. We extend thereby recent results by Fröhlich and Lenzmann, who established in \cite{FL1,FL2} blow-up for solutions to the pseudo-relativistic Hartree equation. As key ingredient for handling the exchange term we use the conservation of the expectation of the square of the angular momentum operator.

math-ph↗
arXiv2007

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 the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N-body Schroedinger equation and the orthogonal projection onto the solution of the Hartree equation.

math-ph↗