  In this paper, we investigate the behavior of the eigenvalues of a magnetic
Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary
conditions in a bounded planar domain. We establish a sharp relation between
the rate of convergence of the eigenvalues as the singular pole is approaching
a boundary point and the number of nodal lines of the eigenfunction of the
limiting problem, i.e. of the Dirichlet Laplacian, ending at that point. The
proof relies on the construction of a limit profile depending on the direction
along which the pole is moving, and on an Almgren-type monotonicity argument
for magnetic operators.
