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Laszlo Erdos

Publications and source records attributed to Laszlo Erdos.

86 records · Page 5Linked to original sources
arXiv2006

Towards the quantum Brownian motion

We consider random Schrödinger equations on $\bR^d$ or $\bZ^d$ for $d\ge 3$ with uncorrelated, identically distributed random potential. Denote by $λ$ the coupling constant and $ψ_t$ the solution with initial data $ψ_0$. Suppose that the space and time variables scale as $x\sim λ^{-2 -κ/2}, t \sim λ^{-2 -κ}$ with $0< κ\leq κ_0$, where $κ_0$ is a sufficiently small universal constant. We prove that the expectation value of the Wigner distribution of $ψ_t$, $\bE W_{ψ_{t}} (x, v)$, converges weakly to a solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data in the weak coupling limit $λ\to 0$. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum $v$.

math-ph↗
arXiv2005

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 $N^2V(N(x_i-x_j))$, where $\bx=(x_1, ..., x_N)$ denotes the positions of the particles. Suppose that the initial data $ψ_{N,0}$ satisfies the condition \[ < ψ_{N,0}, H_N^2 ψ_{N,0} > \leq C N^2 \] where $H_N$ is the Hamiltonian of the Bose system. This condition is satisfied if $ψ_{N,0}= W_N ϕ_{N,0}$ where $W_N$ is an approximate ground state to $H_N$ and $ϕ_{N,0}$ is regular. Let $ψ_{N,t}$ denote the solution to the Schrödinger equation with Hamiltonian $H_N$. Gross and Pitaevskii proposed to model the dynamics of such system by a nonlinear Schrödinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if $u_t$ solves the GP equation, then the family of $k$-particle density matrices $\{\otimes_k u_t, k\ge 1 \}$ solves the GP hierarchy. We prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of $ψ_{N,t}$ are solutions of the GP hierarchy. The uniqueness of the solutions to this hierarchy remains an open question. Our analysis requires that the $N$ boson dynamics is described by a modified Hamiltonian which cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical inter-particle distance. Our proof can be extended to a modified Hamiltonian which only forbids at least $n$ particles from coming close together, for any fixed $n$.

math-ph↗
arXiv2004

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 $N^{-1} V_a(x_i-x_j)$ where $V_a (x) = a^{-3} V (x/a)$. We denote the solution to the $N$-particle Schrödinger equation by $ψ_{N, t}$. Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if $u_t$ solves the GP equation, then the family of $k$-particle density matrices $\{\otimes_k u_t, k\ge 1 \}$ solves the GP hierarchy. Under the assumption that $a = N^{-\eps}$ for $0 < \eps < 3/5$, we prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of $ψ_{N,t}$ are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by $\int V(x) dx$. The uniqueness of the solutions to this hierarchy remains an open question.

math-ph↗
arXiv2004

Uniform Lieb-Thirring inequality for the three dimensional Pauli operator with a strong non-homogeneous magnetic field

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. A new Lieb-Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.

math-ph↗
arXiv2003

Nonlinear Hartree equation as the mean field limit of weakly coupled fermions

We consider a system of N weakly interacting fermions with a real analytic pair interaction. We prove that for a general class of initial data there exists a fixed time T such that the difference between the one particle density matrix of this system and the solution of the non-linear Hartree equation is of order 1/N for any time t less or equal T.

math-ph↗
arXiv2003

Magnetic Lieb-Thirring inequalities with optimal dependence on the field strength

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. Bounds on the sum of the negative eigenvalues are called magnetic Lieb-Thirring (MLT) inequalities. The purpose of this paper is twofold. First, we prove a new MLT inequality in a simple way. Second, we give a short summary of our recent proof of a more refined MLT inequality \cite{ES-IV} and we explain the differences between the two results and methods. The main feature of both estimates, compared to earlier results, is that in the large field regime they grow with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain optimal upper bounds on the pointwise density of zero energy eigenfunctions of the Dirac operator.

math-ph↗
arXiv2003

On the quantum Boltzmann equation

We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrodinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property called restricted quasifreeness in the weak coupling limit at any later time. By definition, a state is called restricted quasifree if the four-point and the eight-point functions of the state factorize in the same manner as in a quasifree state.

math-ph↗
arXiv2002

Derivation of the nonlinear Schrödinger 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 the correlation functions factorize in the limit $N \to \infty$. Furthermore, the limiting one particle density matrix satisfies the nonlinear Hartree equation. The key ingredients are the uniqueness of the BBGKY hierarchy for the correlation functions and a new apriori estimate for the many-body Schrödinger equations.

math-ph↗
arXiv2001

Pauli operator and Aharonov Casher theorem for measure valued magnetic fields

We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual $\bA\in L^2_{loc}$ condition on the vector potential which does not allow to consider such singular fields. We extend the Aharonov-Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted $L^2$ estimate on a singular integral operator.

math-ph↗
arXiv2001

Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field

We consider the long time evolution of a quantum particle weakly interacting with a phonon field. We show that in the weak coupling limit the Wigner distribution of the electron density matrix converges to the solution of the linear Boltzmann equation globally in time. The collision kernel is identified as the sum of an emission and an absorption term that depend on the equilibrium distribution of the free phonon modes.

math-ph↗
arXiv2000

Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case

We establish the exact low-energy asymptotics of the integrated density of states (Lifschitz tail) in a homogeneous magnetic field and Poissonian impurities with a repulsive single-site potential of Gaussian decay. It has been known that the Gaussian potential tail discriminates between the so-called "classical" and "quantum" regimes, and precise asymptotics are known in these cases. For the borderline case, the coexistence of the classical and quantum regimes was conjectured. Here we settle this last remaining open case to complete the full picture of the magnetic Lifschitz tails.

math-ph↗
arXiv2000

The kernel of Dirac operators on $§^3$ and $\R^3$

In this paper we describe an intrinsically geometric way of producing magnetic fields on $§^3$ and $\R^3$ for which the corresponding Dirac operators have a non-trivial kernel. In many cases we are able to compute the dimension of the kernel. In particular we can give examples where the kernel has any given dimension. This generalizes the examples of Loss and Yau (Commun. Math. Phys. 104 (1986) 283-290).

math-ph↗