Absence of embedded eigenvalues for translationally invariant magnetic Laplacians

Abstract

Translationnally invariant bidimensional magnetic Laplacians are considered. Using an improved version of the harmonic approximation, we establish the absence of point spectrum under various assumptions on the behavior of the magnetic field.

Magnetic quantum currents in the presence of a Neumann wall

“…in (1.1), we notice that the operator L h is unitarily equivalent to 2 , where B = h −1 , which shows that the semiclassical limit corresponds to the large magnetic field regime. Such partially semiclassical scaling has already been used by numerous authors in spectral theory of magnetic Schrödinger operators, as, e.g., in [3,4] and in [25], where magnetic models à la Iwatsuka (see [20]) are considered. This paper is also fitted in the semiclassical framework in order to give a lighter formulation of the results and to use the power of the theory of pseudo-differential operators.…”

“…In this case, the existence of edge currents can be linked to the absolutely continuous spectral nature of the spectrum through the use of Mourre's positive commutator method, see, e.g. [5,6], or also [25] (where the absolute continuity is established by elementary means for Iwatsuka Hamiltonians). Similarly, limiting absorption for the quantum Hall Hamiltonian in R 2 + at all energies except Landau levels, was derived in [23] from the monotonicity of the dispersion relations.…”