MAXIMAL SPEED OF QUANTUM PROPAGATION FOR THE HARTREE EQUATION

We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to time-decaying tails, solutions starting in this set stay within the light cone of the corresponding initial datum. We quantify precisely the light cone speed, and hence the speed of nonlinear propagation, in terms of the momentum of the initial state. 1. The Problem and Results In contrast to the key principle of relativity, in Quantum Mechanics, a local change of initial conditions effects the solutions everywhere instantaneously. As Quantum Mechanics is the theory of the quantum matter in non-extreme conditions, it is important to understand the limitations this imposes and to investigate universal properties of the Quantum Mechanical (QM) propagation in some depth. QM predictions are probabilistic and therefore so is the characterization of quantum evolution. It was shown in [35] that, assuming the energy is bounded initially, the supports of solutions of the Schrödinger equation, up to vanishing in time probability tails, spread with a finite speed. This result was improved in [20, 37, 1] and extended in [4] to photon interacting with an atomic or molecular system, while [13, 14] developed a related approach in condensed matter physics. For an electron in an atom the maximal velocity of propagation obtained in [1] is close to the one computed heuristically and to the one observed: of the order of 10m/s which is less than 0.1% of the speed of light. Hence, in a few body case, the QM predictions here are fairly reliable. Though bounds obtained in [35, 20, 37] and especially in [1] are valid for rather general class of potentials, for many-particles systems, their dependence on the number of particles (which in applications could vary from a few to 10) is rather poor. In this paper, we prove bounds on the speed of propagation for bosonic many-body systems in the mean-field approximation, i.e. for the Hartree equation (HE), i ∂ ∂t ψt = (− 1 2 ∆ + V )ψt + v ∗ |ψt|ψt, (1.1) in the physical space R, d ≥ 3. Here V and v are real functions, external (‘onebody’) and internal (interparticle, or ‘pair’) potentials. We assume that the pair 1 2 J. ARBUNICH, J. FAUPIN, F. PUSATERI, AND I. M. SIGAL (convolution) potential v : R → R satisfies v ∈ W γ,(q,∞) if 1 < q < d/2, v ∈ W γ,1 if q = 1, for d/(2q) < γ, (1.2) where W γ,q is the standard Sobolev space (W γ,2 ≡ H), and W γ,(q,∞) denotes the Lorentz-Sobolev space, defined in (1.10) below. Let H := − 2 ∆ + V . We assume also that the external potential V : R → R satisfies (cf. [39, 32]): |〈x〉〈∇〉V | . 1, σ > 5, α ≤ γ, (1.3) H has neither nonpositive eigenvalues nor a resonance at 0. (1.4) We recall that H := − 2 ∆ + V is said to have a resonance at 0 if the equation Hu = 0 has a distributional solution u ∈ 〈x〉L = {f = 〈·〉g | g ∈ L}, for any ν > 1 2 . One can show that there are no zero energy resonances in dimensions d ≥ 5. In Proposition 1.5 below we give explicit restrictions on V for which condition (1.4) is satisfied. Denote χ = χ(r) for a smooth bump function supported in [0, 1] and such that χ = 1 on [0, 1/2]. Let χA denote the characteristic function of a set A. For I ⊂ R a bounded open interval, we define the upper speed (or momentum, as the mass of particles is set equal to 1) bound given that the energy of the initial state ψ0 is supported in I as kI := ‖|∇|χI(H)‖. (1.5) Finally, let B(ε) denote the ball in H centered at the origin and of radius ε > 0. With this, we formulate our main result Theorem 1.1. [Maximal propagation speed for HE] Assume Conditions (1.2), (1.3) and (1.4). Let I be a bounded open interval, g ∈ C∞ 0 (I;R), b > 0 and s ≥ γ. Then, there exist ε > 0 and a set Sg,b ⊂ B(Cε), (1.6) for some absolute C > 0 (see (3.12)), such that the following hold true: (i) Sg,b is in one-to-one correspondence with the set g(H)χ(|x|/b)B(ε); (ii) For any initial condition ψ0 ∈ Sg,b, the Hartree equation (1.1) has a global solution in H and this solution satisfies the estimate ‖χ{|x|≥ct+a}ψt‖ . 〈t〉−1/2, (1.7) for any constants c and a satisfying c > kI and a > b. By definition (3.12) below, the set Sg,b consists of H -functions ‘localized’ in the position x and energy H (essentially to |x| ≤ b and I, respectively). Estimate (1.7) shows that (up to time-decaying tails), a solution to the Hartree equation starting in the set S := ⋃