Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization

Abstract

The theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call "Landau-Weyl pseudodifferential operators". The link between standard Weyl calculus and Landau-Weyl calculus is made explicit by the use of an infinite family of intertwining "windowed wavepacket transforms"; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfunctions of the Hamiltonian operator of a charged particle in a uniform magnetic field, but also to prove global hypoellipticity results and to study the regularity of the solutions to Schrödinger equations.

Fock spaces, Landau operators and the time-harmonic Maxwell equations

“…From the border view of the Heisenberg group, it was recently shown in [13] that the Schrödinger representation and alike play an important role in the study of pseudo-differential operators establishing structural properties between the Weyl calculus and the Landau-Weyl calculus.…”

“…Let us get some motivation from the representation of nilpotent Lie groups (see [29], chapter 10). If one considers operators of the type exp λ n (D − X) as the Clifford extension of the displacement/Heisenberg-Weyl operators (cf [13,36]) to the Heisenberg group H n , we will get an intriguing link that allows us to describe the solutions of the PDE system …”

“…While the study of Landau operators has its roots in the construction of coherent states for relativistic Klein-Gordon and Dirac equations (see [7] and references given there), the surge of interest in these operators arose in the study of the possible occurrence of orbital electromagnetism (cf [19]) as well as in the study of pseudo-differential operators on modulation spaces (cf [13]) and quantum representations of Gabor-windowed Fourier analysis (cf [5]). …”

“…These operators are known in the literature as displacement operators [36] or Heisenberg-Weyl operators [13] underlying the so-called Weyl transform (cf [33], section 1.1).…”

The Schrödinger kernel of the twisted Laplacian and cyclic models

“…The twisted Laplacian appears in harmonic analysis naturally in the context of Wigner transforms and Weyl transforms [14,15], and also in mathematical physics [2][3][4]6]. In particular, it is the Schrödinger operator of a particle moving under the influence of a magnetic field and is of interest in the investigation of the quantum Hall effect.…”