On the eigenvalues of Aharonov-Bohm operators with varying poles: pole approaching the boundary of the domain

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.