I will talk about recent work with Rayan Fahs, Loïc Le Treust, Léo Morin, and Nicolas Raymond.
It concerns the spectral study of purely magnetic Schrödinger operators in dimension 2, in the limit of large fields, which is transformed into a semiclassical limit.
A precise geometric and microlocal analysis (of "normal forms" ) gives a very useful heuristic to reduce the problem to an effective 1D operator.
I will present the case of the confinement of classical and quantum particles by a variable magnetic field, as well as more recent work on the appearance of edge states on bounded domains in the plane, with constant magnetic field.
In both cases we obtain spectral asymptotics with 2 or more terms, for Weyl formulas but also for the precise individual descriptions of a large number of eigenvalues, and their relation with the Landau levels.