I shall introduce an eigenvalue problem for the stationary Aharonov-Bohm Schrödinger operator in the plane. I shall discuss the rate of variation of simple eigenvalues, as the Dirac delta-type
singularity moves, with particular attention to the case of half-integer circulation. The talk collects several results obtained, starting from 2010, in collaboration with L. Abatangelo, V. Bonnaillie-Noël, V. Felli, M. Nys and S. Terracini.