Quantum tunneling with magnetic field in dimension two

Tunneling is a well-known phenomenon in quantum physics: a charged particle in symmetric double-well electric potential is localized in both wells simultaneously. A way to quantify this phenomenon is by estimating the spectral gap between the first two eigenvalues of the corresponding Schrödinger operator. In the semiclassical limit, this gap is exponentially small. Such sharp estimates were obtained in the 80s. However, tunneling with magnetic field turned out to be more tricky, and many questions are still open. I will present recent advances on two dimensional magnetic tunneling. These works exhibit new features which were absent without magnetic field. In particular, exponential decay of eigenfunctions is no longer enough to understand tunneling.