Related papers: On the Cauchy problem for the Hartree approximatio…
We construct generalized quantum Cauchy pre-measures that correspond to the analytic continuation of the transition probability of the Cauchy process to imaginary time. We show that these complex pre-measures of time translations extend to…
This paper is concerned with the hypercoercivity property of solutions to the Cauchy problem on the linear Boltzmann equation with a confining potential force. We obtain the exponential time rate of solutions converging to the steady state…
We consider the Cauchy problem for (energy-subcritical) nonlinear Schr\"odinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum…
Here we derive some results on so called quantitative Runge approximation in the case of the time-harmonic Maxwell equations. This provides a Runge approximation having more explicit quantitative information. We additionally derive some…
The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations is addressed with the method of linearization and fixed point. Coupling between the hyperbolic and parabolic variables is allowed in the linearization and we do…
We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\Delta u+(|\cdot|^{-3}\ast|u|^{2})u=0$ in $\mathbb{R}^{5}$ with the initial data in $H^1$, and study the divergent property of infinite-variance and nonradial…
Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first show that any variational…
We consider the Cauchy problem associated to the fourth-order nonlinear Schr\"{o}dinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise…
The Cauchy problem for the Schr\"odinger equations is studied with time-dependent potentials growing polynomially in the spatial direction. First the existence and the uniqueness of solutions are shown in the weighted Sobolev spaces. In…
This is a work extending the results of \cite{AH} and \cite{AHH}. We want to show convergence of the Schr\"odinger equation towards the Hartree equation with more natural assumptions. We first consider both the defocusing and the focusing…
The non autonomous Cauchy problem for time dependent 1D point interactions is considered. The regularity assumptions for the coupling parameter are accurately analyzed and show that the general results for non autonomous linear evolution…
This paper studies the properties of solutions for a double nonlinear time-dependent parabolic equation with variable density, not in divergence form with a source or absorption. The problem is formulated as a partial differential equation…
Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant…
In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the Coulomb potential 1/|x|, and it produces the long-range interaction…
We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of…
We show how the linear delta expansion, as applied to the slow-roll transition in quantum mechanics, can be recast in the closed time-path formalism. This results in simpler, explicit expressions than were obtained in the Schr\"odinger…
This work studies the Cauchy problem for the energy-critical inhomogeneous Hartree equation with inverse square potential $$i\partial_t u-\mathcal K_\lambda u=\pm |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, \quad \mathcal…
The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in…
We study a Hartree ensemble approximation for real-time dynamics in the toy model of 1+1 dimensional scalar field theory. Damping behavior seen in numerical simulations is compared with analytical predictions based on perturbation theory in…
The time-dependent Hartree and Hartree-Fock equations provide effective mean-field descriptions for the dynamics of large fermionic systems and play a fundamental role in many areas of physics. In this work, we rigorously derive the…