Finite energy subspace for time-periodic Schrödinger operators

TL;DR

The paper addresses the long-time behavior of $N$-body quantum systems in time-periodic short-range fields, introducing a finite energy subspace ${\mathcal H}^{+}_{\mathrm{ener}}$ that geometrically captures states with finite asymptotic energy. Using time-dependent commutator methods inspired by Yafaev, it proves the existence of channel wave operators for all channels and establishes a geometric identification ${\mathcal H}^{+}_{\mathrm{wave}} = {\mathcal H}^{+}_{\mathrm{ener}}$, thereby recasting asymptotic completeness in a finite-energy language. For ${N=2}$, the result yields asymptotic completeness by showing ${\mathcal H}^{+}_{\mathrm{ener}} = {\mathcal H}_{ac}$, while for ${N\ge 3}$ the completeness question remains open; however, the framework provides a concrete intermediate step and a sharp minimal-velocity bound that could drive future progress. The work develops a rich set of tools, including Yafaev-type localization observables, square-root regularization schemes, and radiation-condition bounds, highlighting a deep connection between quantum scattering in time-periodic settings and classical phase-space dynamics. The results apply notably to AC-Stark models (time-periodic, zero-mean external fields) and offer a rigorous geometric perspective on the large-time structure of scattering states in time-periodic N-body systems.

Abstract

For $N$-body Schrö\-dinger operators with time-periodic short-range pair-potentials we show by a time-dependent commutator method that all channel wave operators exist. For $N=2$ we prove asymptotic completeness by a simplified version of the method, recovering Yajima's completeness result \cite{Yaj1} proven by a stationary method. We propose a definition of a \emph{finite energy subspace}, intuitively consisting of states with `finite asymptotic energy'. This geometric notion is used to characterize the \emph{wave operator subspace} given as the direct sum of the ranges of the channel wave operators. Thus our main result states that the two subspaces coincide. In turn they coincide for $N= 2$ with the orthogonal subspace of the pure point subspace of the monodromy operator (by asymptotic completeness). For $N\geq 3$ asymptotic completeness for time-periodic short-range potentials remains an open problem. The main result of the paper may in this case potentially serve as an intermediate step for proving (or possibly disproving) asymptotic completeness. Another potential ingredient could be a key intermediate result of the paper, asserting that the states orthogonal to the pure point subspace obey a (sharp) \emph{minimal velocity bound}. Thus it remains an open problem possibly to derive a good maximal velocity bound for $N\geq 3$. Our results apply to $N$-body systems of particles in a time-periodic external electric field with time-mean equal to zero, for example the AC-Stark model.

Theorems & Definitions (55)

• Theorem 2.2
• Theorem 2.3
• Theorem 2.6
• Theorem 2.9
• Theorem 2.10
• Proposition 2.11
• Proposition 2.12
• Lemma 2.13
• proof
• Proposition 2.14