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 by Helffer-Sjöstrand in the 80’s. When the localization is induced by a magnetic field instead, without electric potential, a similar effect was conjectured by Helffer-Morame in 1996. However, the magnetic case turned out to be more complicated to understand. After recent partial results, we finally proved such a tunneling formula between purely magnetic wells, hence answering a 30 years old problem. Interestingly, the size of the interaction is smaller than the one conjectured in 1996.

As we will see during the talk, we had to revisit the Helffer-Sjöstrand theory, in light of recent advances in the study of magnetic wells. In particular, our proof relies on the understanding of the dynamical properties of magnetic Laplacians. In the semiclassical limit, we manage to split the effects of the cyclotron and center guide motions. It turns out that tunneling occurs along the center guide variables only, the cyclotron motion acting on top of it. A key aspect of our analysis is getting exponential decay estimates of eigenfunctions in phase space, where the position and momenta variables are mixed.

Joint work with Yannick Guedes-Bonthonneau, Søren Fournais and Nicolas Raymond