Maximal speed of quantum propagation for the Hartree equation. (English) Zbl 1529.35459
Summary: 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.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35Q40 | PDEs in connection with quantum mechanics |
| 81V73 | Bosonic systems in quantum theory |
| 81V70 | Many-body theory; quantum Hall effect |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
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