Long-Time Behaviour of Time-Dependent Density Functional Theory

Pusateri, Fabio; Sigal, Israel Michael

doi:10.1007/s00205-021-01656-1

Cited by 8 publications

(6 citation statements)

References 30 publications

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“…The data are taken in entropy classes relative to a referential state corresponding to our equilibria. Recently again, Pusateri and Sigal, [PS21] proved scattering near the 0 solution for the equation i∂ t γ = [−∆ + f (ρ γ ), γ] for a large class of nonlinearities f that include w * ρ γ where w belongs to a weak L r , for r ∈ (1, d). Their result is quite general as it narrowly misses the Coulomb potential.…”

Section: An Effective Equation For Fermionsmentioning

confidence: 99%

Stability of steady states for Hartree and Schrödinger equations for infinitely many particles

Collot¹,

Suzzoni²

2022

Annales Henri Lebesgue

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We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation for density matrices. We treat dimensions 2 and 3, extending our previous result. We reach a large class of interaction potentials, which includes the nonlinear Schrödinger equation. This result has an incidence in the density matrices framework. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger equation, and on the use of explicit low frequency cancellations as done by Lewin and Sabin. To relate to density matrices, we use Strichartz estimates for orthonormal systems from Frank and Sabin, and Leibniz rules for integral operators.

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Section: An Effective Equation For Fermionsmentioning

confidence: 99%

Stability of steady states for Hartree and Schrödinger equations for infinitely many particles

Collot¹,

Suzzoni²

2022

Annales Henri Lebesgue

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“…For example, see [13,Corollary 1]. The author could not find any clear statement in the literature, but (15) immediately implies Proposition 1.5…”

Section: Linear Responsementioning

confidence: 95%

“…The Cauchy and scattering problems for (6) itself have been studied by [2,3,4,17] and more recently, by [15]. (More precisely, they considered more general nonlinear terms.)…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability of a wide class of steady states for the Hartree equation for random fields

Sonae¹

2023

Preprint

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We study the Hartree equation describing the time evolution of the wave functions of infinitely many fermions interacting with each other. The Hartree equation can be formulated in terms of random fields. This formulation was introduced by de Suzzoni in [16]. It has infinitely many steady states, and [7,8] have studied its asymptotic stability. However, they required somewhat strong conditions for steady states, and the stability of Fermi gas at zero temperature, one of the most important steady states from the physics point of view, was left open. In this paper, we prove the stability of steady states in a wide class, which includes Fermi gas at zero temperature in d-dimensional space when d ≥ 3.This paper is the revised version. There are two major changes. The first one is that the assumptions for potential w in Theorem 1.3 are weakened. The second one is that we removed Theorem 1.4 in the previous version because the author found some gaps which seemed difficult to fix.

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“…Mathematically, this is a complicated nonlinear integro-differential equation. Although existence, uniqueness and various properties of this and related equations has been studied mathematically in various regimes [11,15,30,33], the linear response theory and its numerical approximation has not, to the best of our knowledge. The above model is complicated, and for the purposes of mathematical analysis we will consider in this paper the drastic over-simplification V HXC = 0.…”

Section: Introductionmentioning

confidence: 99%

Finite-size effects in response functions of molecular systems

Dupuy¹,

Levitt²

2022

The SMAI Journal of Computational Mathematics

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We consider an electron in a localized potential submitted to a weak external, time-dependent field. In the linear response regime, the response function can be computed using Kubo's formula. In this paper, we consider the numerical approximation of the response function by means of a truncation to a finite region of space. This is necessarily a singular approximation because of the discreteness of the spectrum of the truncated Hamiltonian, and in practice a regularization (smoothing) has to be used. Our results provide error estimates for the response function past the ionization threshold with respect to both the smoothing parameter and the size of the computational domain.

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Long-Time Behaviour of Time-Dependent Density Functional Theory

Cited by 8 publications

References 30 publications

Stability of steady states for Hartree and Schrödinger equations for infinitely many particles

Stability of steady states for Hartree and Schrödinger equations for infinitely many particles

Asymptotic stability of a wide class of steady states for the Hartree equation for random fields

Finite-size effects in response functions of molecular systems

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