1,720,986 research outputs found
Dynamical Collapse of Boson Stars
No full textWe study the time evolution in a system of N bosons with a relativistic dispersion law interacting through a Newtonian gravitational potential with coupling constant G. We consider the mean field scaling where N tends to infinity, G tends to zero and λ = GN remains fixed. We investigate the relation between the many body quantum dynamics governed by the Schrödinger equation and the effective evolution described by a (semi-relativistic) Hartree equation. In particular, we are interested in the super-critical regime of large λ [the sub-critical case has been studied in Elgart and Schlein (Comm Pure Appl Math 60(4):500-545, 2007) and Knowles and Pickl (Commun Math Phys 298(1):101-138, 2010)], where the nonlinear Hartree equation is known to have solutions which blow up in finite time. To inspect this regime, we need to regularize the interaction in the many body Hamiltonian with an N dependent cutoff that vanishes in the limit N → ∞. We show, first, that if the solution of the nonlinear equation does not blow up in the time interval [-T, T], then the many body Schrödinger dynamics (on the level of the reduced density matrices) can be approximated by the nonlinear Hartree dynamics, just as in the sub-critical regime. Moreover, we prove that if the solution of the nonlinear Hartree equation blows up at time T (in the sense that the H 1/2 norm of the solution diverges as time approaches T), then also the solution of the linear Schrödinger equation collapses (in the sense that the kinetic energy per particle diverges) if t → T and, simultaneously, N → ∞ sufficiently fast. This gives the first dynamical description of the phenomenon of gravitational collapse as observed directly on the many body level. © 2011 Springer-Verlag
A new second-order upper bound for the ground state energy of dilute Bose gases
Full text linkWe establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee-Huang-Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported, gives better rates and, in our opinion, is substantially simpler
A Second Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime
Full text linkWe prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius a/ N, moving in the three-dimensional unit torus Λ. Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit N→ ∞. The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales
Ground state energy of a Bose gas in the Gross-Pitaevskii regime
Full text linkWe review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson's upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross-Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross-Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere
Quantum Many-Body Fluctuations Around Nonlinear Schrödinger Dynamics
Full text linkWe consider the many-body quantum dynamics of systems of bosons interacting through a two-body potential N^(3b-1)V(N^bx) , scaling with the number of particles N. For b in the interval (0,1) , we obtain a norm-approximation of the evolution of an appropriate class of data on the Fock space. To this end, we need to correct the evolution of the condensate described by the one-particle nonlinear Schrödinger equation by means of a fluctuation dynamics, governed by a quadratic generator
Dynamical formation of correlations in a Bose-Einstein condensate
No full textWe 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 2 V(N(x i − x j )). We monitor the behaviour 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
Mean field evolution of fermions with Coulomb interaction
Full text linkWe study the many body Schrödinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree–Fock dynamics. Our result holds under a condition on the solution of the Hartree–Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree–Fock evolution is trivial), but that we expect to hold more generally
The excitation spectrum of Bose gases interacting through singular potentials
Full text linkWe consider systems of N bosons in a box of volume 1, interacting through a repulsive two-body potential of the form kN^(3b−1)V(Nbx). For all b in the interval (0,1), and for sufficiently small coupling constant k positive, we establish the validity of Bogolyubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large N
Effective Dynamics of Extended Fermi Gases in the High-Density Regime
Full text linkWe study the quantum evolution of many-body Fermi gases in three dimensions,
in arbitrarily large domains. We consider both particles with non-relativistic
and with relativistic dispersion. We focus on the high-density regime, in the
semiclassical scaling, and we consider a class of initial data describing
zero-temperature states. In the non-relativistic case we prove that, as the
density goes to infinity, the many-body evolution of the reduced one-particle
density matrix converges to the solution of the time-dependent Hartree
equation, for short macroscopic times. In the case of relativistic dispersion,
we show convergence of the many-body evolution to the relativistic Hartree
equation for all macroscopic times. With respect to previous work, the rate of
convergence does not depend on the total number of particles, but only on the
density: in particular, our result allows us to study the quantum dynamics of
extensive many-body Fermi gases.Comment: 44 pages, no figures. The proof of the main result has been
simplified and generalized. Furthermore, we included the derivation of the
time-dependent relativistic Hartree equation, for all macroscopic time
Effective evolution equations from quantum dynamics
Full text linkThese notes investigate the time evolution of quantum systems, and in particular the rigorous derivation of effective equations approximating the many-body Schrödinger dynamics in certain physically interesting regimes. The focus is primarily on the derivation of time-dependent effective theories (non-equilibrium question) approximating many-body quantum dynamics. The book is divided into seven sections, the first of which briefly reviews the main properties of many-body quantum systems and their time evolution. Section 2 introduces the mean-field regime for bosonic systems and explains how the many-body dynamics can be approximated in this limit using the Hartree equation. Section 3 presents a method, based on the use of coherent states, for rigorously proving the convergence towards the Hartree dynamics, while the fluctuations around the Hartree equation are considered in Section 4. Section 5 focuses on a discussion of a more subtle regime, in which the many-body evolution can be approximated by means of the nonlinear Gross-Pitaevskii equation. Section 6 addresses fermionic systems (characterized by antisymmetric wave functions); here, the fermionic mean-field regime is naturally linked with a semiclassical regime, and it is proven that the evolution of approximate Slater determinants can be approximated using the nonlinear Hartree-Fock equation. In closing, Section 7 reexamines the same fermionic mean-field regime, but with a focus on mixed quasi-free initial data approximating thermal states at positive temperature
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