Dynamical Collapse of Boson Stars

Abstract

We study the time evolution in system of $N$ bosons with a relativistic dispersion law interacting through an attractive Coulomb potential with coupling constant $G$. We consider the mean field scaling where $N$ tends to infinity, $G$ tends to zero and $\lambda =GN$ 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 $\lambda$ (the sub-critical case has been studied in \cite{ES,KP}), 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 Coulomb interaction in the many body Hamiltonian with an $N$ dependent cutoff that vanishes in the limit $N\to \infty$. 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 $t\to T$ and, simultaneously, $N\to \infty$ sufficiently fast. This gives the first dynamical description of the phenomenon of gravitational collapse as observed directly on the many body level.

40 pages

References in corpus (4)

• Rigorous Derivation of the Gross-Pitaevskii Equation
• Mean-Field Dynamics: Singular Potentials and Rate of Convergence
• Second-order corrections to mean field evolution for weakly interacting Bosons. I
• Quantum Dynamics with Mean Field Interactions: a New Approach

Cited by in corpus (26)

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• Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation
• On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy
• Mean-Field Dynamics of Fermions with Relativistic Dispersion
• Bogoliubov corrections and trace norm convergence for the Hartree dynamics
• The Boson star equation with initial data of low regularity
• On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics
• A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on ${\mathbb{T}}^3$ from the dynamics of many-body quantum systems
• Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy
• Modified scattering for the Boson Star Equation
• The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
• The Derivation of the ${\mathbb{T}}^3$ Energy-critical NLS from Quantum Many-body Dynamics
• Derivation of the Hartree equation for compound Bose gases in the mean field limit
• Randomization and the Gross-Pitaevskii hierarchy
• A simple proof of convergence to the Hartree dynamics in Sobolev trace norms
• Small data scattering for semi-relativistic equations with Hartree type nonlinearity
• Mean-field quantum dynamics with magnetic fields
• Rate of Convergence Towards Semi-Relativistic Hartree Dynamics
• Local existence of solutions to Randomized Gross-Pitaevskii hierarchies
• Wave dispersion derived from the square-root Klein-Gordon-Poisson system
• Rate of Convergence towards Hartree Dynamics with Singular Interaction Potential
• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• Mean-field limits of particles in interaction with quantized radiation fields
• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
• Many-Body Blow-Up Profile of Boson Stars with External Potentials
• Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential