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Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field

Dario Bambusi, Benoit Grébert, Alberto Maspero, Didier Robert, Carlos Villegas-Blas

TL;DR

The paper analyzes time-quasiperiodic perturbations of the 2D Landau Hamiltonian under ${\mathbf B}(t)=B_0+\varepsilon f(\omega t)$ and reveals a gauge-dependent dichotomy: in the Landau gauge the dynamics are generically unbounded while the Floquet spectrum is absolutely continuous, whereas in the symmetric gauge the system is reducible to two decoupled harmonic oscillators, yielding bounded dynamics and a discrete Floquet spectrum. The authors develop a rigorous KAM reducibility framework in an extended phase space to construct near-identity symplectic changes that produce simple quadratic normal forms for a full-measure set of nonresonant frequencies $\omega$, and they translate these classical results into quantum consequences for the propagator and spectrum. Central to the approach are homological equations, iterative KAM steps, and careful control of small divisors, including adaptations when resonances appear. The results underscore how gauge choice, even with the same magnetic field, fundamentally affects long-time behavior in time-dependent quantum systems and provide a robust mechanism for obtaining reducibility and spectral conclusions in quadratic Hamiltonian families.

Abstract

We consider a modulated magnetic field, $B(t) = B_0 +\varepsilon f(ωt)$, perpendicular to a fixed plane, where $B_0$ is constant, $\varepsilon>0$ and $f$ a periodic function on the torus ${\mathbb T}^n$. Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for "almost all" non resonant frequencies $ω$. On the contrary, for the symmetric gauge we obtain that, for "almost all" non resonant frequencies $ω$, the Landau Hamiltonian is reducible to a two dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.

Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field

TL;DR

The paper analyzes time-quasiperiodic perturbations of the 2D Landau Hamiltonian under and reveals a gauge-dependent dichotomy: in the Landau gauge the dynamics are generically unbounded while the Floquet spectrum is absolutely continuous, whereas in the symmetric gauge the system is reducible to two decoupled harmonic oscillators, yielding bounded dynamics and a discrete Floquet spectrum. The authors develop a rigorous KAM reducibility framework in an extended phase space to construct near-identity symplectic changes that produce simple quadratic normal forms for a full-measure set of nonresonant frequencies , and they translate these classical results into quantum consequences for the propagator and spectrum. Central to the approach are homological equations, iterative KAM steps, and careful control of small divisors, including adaptations when resonances appear. The results underscore how gauge choice, even with the same magnetic field, fundamentally affects long-time behavior in time-dependent quantum systems and provide a robust mechanism for obtaining reducibility and spectral conclusions in quadratic Hamiltonian families.

Abstract

We consider a modulated magnetic field, , perpendicular to a fixed plane, where is constant, and a periodic function on the torus . Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for "almost all" non resonant frequencies . On the contrary, for the symmetric gauge we obtain that, for "almost all" non resonant frequencies , the Landau Hamiltonian is reducible to a two dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.
Paper Structure (14 sections, 14 theorems, 163 equations)

This paper contains 14 sections, 14 theorems, 163 equations.

Table of Contents

  1. Introduction and main results
  2. Main result in the Landau gauge
  3. Main result in symmetric gauge
  4. About the change of gauge
  5. Related literature
  6. Proofs of main Theorems
  7. Proof of Theorem \ref{['len.mag']}
  8. Proof of Theorem \ref{['len.mag.sym']}
  9. Proofs of two reducibility Theorems
  10. General strategy
  11. Homological equation
  12. Iterative lemma
  13. Transition to the limit and proof of Theorem \ref{['KAMclassico']}
  14. Changes for proving Theorem \ref{['KAMclassico1']}.

Key Result

Theorem 1.2

There exists $\varepsilon_0>0$ such that for $|\varepsilon|<\varepsilon_0$, there exist such that, for $\omega\in\mathcal{C}_\varepsilon$, the time quasi-periodic Hamiltonian $h_L(t,x,p)$ in landaub is conjugated to the constant coefficient quadratic Hamiltonian where and $c_\omega$ is given by c.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • proof
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • ...and 12 more