Hartree-Refreshed Continuum Density-Selected Phase Drives in the Regular Mean-Field Regime

Panasenko

Published Apr 26, 2026• 0 citations

Abstract

We formulate a regular-regime theorem for continuum density-selected phase drives refreshed along a Hartree reference path. In contrast to a prescribed continuum drive, the density path is generated by the reference Hartree solution itself, namely ρt = |φt|2 . The microscopic comparison is nevertheless made against the corresponding time-dependent Hartree-refreshed reference field, not against a fully microscopic feedback law depending on the evolving reduced density of the same N-body state. In this regular finite-window setting we isolate three mechanisms: construction of the Hartree-refreshed reference branch, convergence of finite refreshed phase schedules to an abelian multiplication product integral, and a Pickl-type propagation-of-condensation estimate for the associated many-body reference dynamics.

DOI: 10.5281/zenodo.19800106

Related papers

Short Hartree-Refreshed Pulses: Endpoint Transfer and Post-Pulse Mean-Field Closure

Panasenko

This preprint is Paper IV in the research series “Operator Properties and Mathematical Structure of Nonlocal Morphing”. We study short pulse windows in which the phase generator is refreshed along a self-consistent Hartree reference path and the microscopic pulse is driven by the corresponding Hartree-refreshed field lifted to the many-body level. On the pulse window we construct the refreshed reference branch by a local Picard argument, compare the refreshed and frozen microscopic pulse propagators by Duhamel’s formula, and transfer the resulting mismatch to an explicit one-body endpoint defect. We then formulate an abstract restart mechanism showing that any endpoint defect controlled at time τ can be propagated by the standard post-pulse mean-field closure. In the theorem-clean regular regime, and on every fixed post-pulse window [τ, τ+T], we prove that the one-body defect is bounded by C_T·(τ + N^{-1}), provided the frozen regular-regime endpoint scale is available. The main new contribution is the endpoint-transfer mechanism for short Hartree-refreshed pulses: the pulse-window discrepancy is converted into an explicit one-body endpoint defect, and the post-pulse evolution is then handled by a restarted mean-field closure. Thus the frozen bridge remains the leading-order law for short Hartree-refreshed pulses, while the refreshed pulse enters the post-pulse theory only through an explicit endpoint defect. This record is published as a citable preprint in order to provide a public reference point for the companion mean-field arguments used in the present paper and to make the manuscript openly accessible prior to journal submission.

Apr 12, 20260 citations

Propagation of Chaos after a Frozen Density-Selected Phase Imprint in the Bosonic Mean-Field Limit

Panasenko

This record contains the manuscript “Propagation of Chaos after a Frozen Density-Selected Phase Imprint”. Abstract. We study a bosonic mean-field system after a single frozen density-selected phase imprint, with phase profile determined by the pre-imprint density. The main result isolates the exact frozen-density interface at the imprint time and shows that the one-particle defect is transported through the jump without additional loss. On each fixed observation window, this reduces the post-imprint analysis to the standard Hartree propagation-of-chaos framework and yields the usual O(N^{-1}) trace-norm convergence in the regular bounded-integrable kernel regime. The manuscript also records two controlled extensions. First, finite-duration imprints are compared with the same frozen benchmark through area matching, with the dynamic-reference mismatch quantified separately. Second, for prescribed finite kick schedules independent of N, the same bookkeeping yields a conservative fixed-schedule extension on fixed windows. In addition, a hypothesis-dependent weakly singular branch is included under explicit extra spacetime assumptions; this branch is conditional and is stated separately from the unconditional regular result. The paper is intended as a rigorous bridge result: it identifies the exact frozen-density jump law, the induced defect transport, and the subsequent standard Hartree comparison. It does not claim a refreshed-generator theorem, a continuum-drive limit, or a uniform-in-the-number-of-kicks result. Files in this record include the main manuscript and supporting source materials.

Apr 20, 20260 citations

Mean-Field Limit for Bosonic Systems with Microscopic Density-Dependent Phase Drives

Panasenko

Mean-field limits for bosonic systems explain how an effective one-particle nonlinear equationemerges from an N-body Schrödinger dynamics. In the standard mean-field setting, themicroscopic Hamiltonian contains a scaled pair interaction, and the limiting dynamics is aHartree equation. This paper considers a closely related but structurally different situation: inaddition to the usual pair interaction, the microscopic system carries a phase drive which isgenerated by the evolving N-body state itself.The distinction is important. The phase field considered here is not a prescribed timedependent potential and is not imported from the limiting Hartree trajectory. At finite N, it iscomputed from the normalized one-particle marginal density

Apr 28, 20260 citations

Frozen Density-Selected Unitaries: Exact Action on Marginals, Operator Regularity, and Controlled Singular-Kernel Extensions

Panasenko

This record contains the manuscript “Frozen Density-Selected Unitaries: Exact Action on Marginals, Operator Regularity, and Controlled Singular-Kernel Extensions”. The paper studies a one-step class of density-selected phase transformations in bosonic quantum dynamics. The transformation is frozen in the sense that the density profile used to generate the phase is fixed externally during the imprint, rather than being updated through a self-consistent feedback loop. The manuscript develops three complementary results. First, it proves the exact action of the frozen product-imprint map on reduced marginals of bosonic many-body states. Second, it establishes regularity properties of the associated operator map in the bounded-kernel regime, including Lipschitz continuity and differentiability with respect to the prescribed input density. Third, it gives a controlled extension to singular kernels, where operator-norm convergence is generally too strong and strong-operator convergence becomes the natural topology. The paper is intended as an operator-theoretic foundation for density-selected phase-imprint models and as a bridge toward later mean-field and Hartree-type analyses. It separates the frozen external-density layer from genuinely self-consistent feedback mechanisms and clarifies which conclusions belong to each regime.

Apr 11, 20260 citations

A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics

Sören Petrat, Peter Pickl

We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151–164 2011 ) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.

arXiv:1409.0480Mathematical Physics Analysis and GeometryFeb 16, 201657 citations

A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

Ioannis Anapolitanos, Michael Hott

The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the kth marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see the work of Michelangeli and Schlein [Commun. Math. Phys. 311(3), 645–687 (2012)] and Lührmann [J. Math. Phys. 53(2), 022105 (2012)]. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional aN introduced by Pickl [Lett. Math. Phys. 97(2), 151–164 (2011)].

arXiv:1608.01192Journal of Mathematical PhysicsDec 1, 201618 citations