This talk is devoted to the spectrum of the electro-magnetic Laplacian $(-ih\nabla-A)^2-hB$ on a bounded, regular, and simply-connected open set of the plane.

Here $B$ is the magnetic field associated with $A$. We will see that, when the magnetic field is positive (and under generic conditions), we can accurately describe the low-lying eigenvalues in the semiclassical limit $h\to 0$. We will show the crucial role of the magnetic Cauchy-Riemann operators (and of their ellipticity), of the Bergman-Hardy spaces, and of the Riemann mapping theorem in the description of the spectrum.

This is a joint work with Jean-Marie Barbaroux, Loïc Le Treust, and Edgardo Stockmeyer.