  We continue the analysis started in [Noris,Terracini,Indiana Univ Math
J,2010] and [Bonnaillie-No\"el,Noris,Nys,Terracini,Analysis & PDE,2014],
concerning the behavior of the eigenvalues of a magnetic Schr\"odinger operator
of Aharonov-Bohm type with half-integer circulation. We consider a planar
domain with Dirichlet boundary conditions and we concentrate on the case when
the singular pole approaches the boundary of the domain. The $k$-th magnetic
eigenvalue converges to the $k$-th eigenvalue of the Laplacian. We can predict
both the rate of convergence and whether the convergence happens from above or
from below, in relation with the number of nodal lines of the $k$-th
eigenfunction of the Laplacian. The proof relies on a detailed asymptotic
analysis of the eigenfunctions, based on an Almgren-type frequency formula for
magnetic eigenfunctions and on the blow-up technique.
