The paper studies the dynamics of non-autonomous Schrödinger equations on lattices, where the Hamiltonian is given by H(t) = H0 + V(t) with H0 a long-range self-adjoint operator satisfying a power-law decay condition.
The key results are:
Theorem 1.1 provides sub-ballistic upper bounds on the outside probabilities P(N,t), i.e., the probability that the wavepacket is found outside a region |x| ≥ Ntα, for 1/2 < α ≤ 1. The main assumption is that the momentum vanishes sufficiently fast in the propagation front.
Theorem 1.2 establishes a general ballistic upper bound on P(N,t), independent of the solution. This extends the classical Radin-Simon bound to the non-autonomous setting.
Theorem 1.3 shows that higher moments of the position operator are also confined to the linear light cone, provided the initial state has bounded higher moments.
Theorem 1.4 extends these results to generic non-linear, non-autonomous Schrödinger equations on lattices via a frozen-coefficient argument.
The proofs rely on commutator estimates and the construction of suitable propagation observables that track the spacetime localization of the evolving wavepackets. The results apply to a wide class of long-range Hamiltonians and can be useful for understanding the dynamics of quantum many-body systems at the mean-field level.
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by Jingxuan Zha... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2409.13762.pdfDeeper Inquiries