  We consider a magnetic operator of Aharonov-Bohm type with Dirichlet boundary
conditions in a planar domain. We analyse the behavior of its eigenvalues as
the singular pole moves in the domain. For any value of the circulation we
prove that the k-th magnetic eigenvalue converges to the k-th eigenvalue of the
Laplacian as the pole approaches the boundary. We show that the magnetic
eigenvalues depend in a smooth way on the position of the pole, as long as they
remain simple. In case of half-integer circulation, we show that the rate of
convergence depends on the number of nodal lines of the corresponding magnetic
eigenfunction. In addition, we provide several numerical simulations both on
the circular sector and on the square, which find a perfect theoretical
justification within our main results, together with the ones in
[Bonnaillie-No\"el, V., Helffer, B., Exp. Math. 20 (2011), no. 3, 304-322;
MR2836255 (2012i:35274)].
