The mathematical analysis of the tunnelling effect between electric wells was a key milestone in semiclassical analysis reached in the 1980’s with important contributions by Simon and Helffer-Sjöstrand. However, the analysis of the effect in the presence of magnetic fields was largely left open. Technically, this is because the complex phases that a magnetic field induces in the eigenfunctions causes cancellations that are delicate to control to the required precision. Although a general theory of magnetic tunnelling is still lacking at the level of generality of the non-magnetic theory, there has been important progress in recent years. I will review some recent results, both in very special geometries where separation of variables makes analysis more manageable, and a recent result which is the first “generic” calculation of a purely magnetic tunnelling effect.
Results are in particular in collaboration with Yannick Guedes Bonthonneau, Leo Morin and Nicolas Raymond.