Geodesic
Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field
Dario Bambusi1 orcid ; Benoit Grébert2 orcid ; Alberto Maspero3 orcid ; Didier Robert2 orcid ; Carlos Villegas-Blas4 orcid
1Università degli Studi di Milano, Milan, Italy
2Université de Nantes, Nantes, France
3SISSA, Trieste, Italy
4Universidad Nacional Autonoma de México, Ciudad de Mexico, Mexico
EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 155-184

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We consider a modulated magnetic field, B(t)=B0​+εf(ωt), perpendicular to a fixed plane, where B0​ is constant, ε>0 and f a periodic function on the torus Tn. 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.
DOI : 10.4171/emss/99
Classification : 37K55, 35Q56
Mots-clés : Landau Hamiltonian, KAM theory, growth of Sobolev norms

Dario Bambusi  1   ; Benoit Grébert  2   ; Alberto Maspero  3   ; Didier Robert  2   ; Carlos Villegas-Blas  4

1 Università degli Studi di Milano, Milan, Italy
2 Université de Nantes, Nantes, France
3 SISSA, Trieste, Italy
4 Universidad Nacional Autonoma de México, Ciudad de Mexico, Mexico 
Dario Bambusi; Benoit Grébert; Alberto Maspero; Didier Robert; Carlos Villegas-Blas. Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field. EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 155-184. doi: 10.4171/emss/99
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     title = {Longtime dynamics for the {Landau} {Hamiltonian} with a~time dependent magnetic field},
     journal = {EMS surveys in mathematical sciences},
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