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Slow dispersion in Floquet-Dirac Hamiltonians

Anthony Bloch, Amir Sagiv, Stefan Steinerberger

Abstract

We study dispersive decay for non-autonomous Hamiltonian systems. While the general theory for dispersion in such non-autonomous systems is largely open, it was shown \cite{kraisler2025time} that there exists a time-periodically forced one-dimensional Dirac equation with unusually slow dispersive decay rate of $t^{-1/5}$. It is to be expected that such behavior is not generic and requires a very particular forcing term; we provide a more general ansatz and systematic procedure to construct such an equation with a dispersive decay rate no faster than $t^{-1/10}$. Our limitations are purely algebraic and it stands to reason that arbitrarily slow decay, $t^{-\varepsilon}$ for every $\varepsilon > 0$, should be achievable.

Slow dispersion in Floquet-Dirac Hamiltonians

Abstract

We study dispersive decay for non-autonomous Hamiltonian systems. While the general theory for dispersion in such non-autonomous systems is largely open, it was shown \cite{kraisler2025time} that there exists a time-periodically forced one-dimensional Dirac equation with unusually slow dispersive decay rate of . It is to be expected that such behavior is not generic and requires a very particular forcing term; we provide a more general ansatz and systematic procedure to construct such an equation with a dispersive decay rate no faster than . Our limitations are purely algebraic and it stands to reason that arbitrarily slow decay, for every , should be achievable.

Paper Structure

This paper contains 8 sections, 9 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Decay estimates.
  3. A non-autonomous system.
  4. Obtaining dispersive decay estimates
  5. Existence of solutions
  6. Preliminaries
  7. The Construction
  8. Proof of Lemma \ref{['lem:akszero']} - existence of roots

Key Result

Theorem 1

There exists a function $\nu(t)$ which assumes the values $m \sigma_1$ and $-m \sigma_1$ periodically in time so that the (generic) dispersive decay rate is no faster than $t^{-1/3}$. Moreover, there exist (nongeneric) choices of parameters where it is no faster than $t^{-1/5}$.

Theorems & Definitions (17)

  • Theorem : informally, Kraisler, Sagiv, Weinstein kraisler2025time
  • Theorem 1.1: Main Result
  • Conjecture
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm:exp']}
  • proof : proof of Theorem \ref{['thm:main']}
  • Remark 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 7 more