  We show that the presence of negative eigenvalues in the spectrum of the
angular component of an electromagnetic Schr\"odinger hamiltonian $H$
generically produces a lack of the classical time-decay for the associated
Schr\"odinger flow $e^{-itH}$. This is in contrast with the fact that
dispersive estimates (Strichartz) still hold, in general, also in this case. We
also observe an improvement of the decay for higher positive modes, showing
that the time decay of the solution is due to the first nonzero term in the
expansion of the initial datum as a series of eigenfunctions of a quantum
harmonic oscillator with a singular potential. A completely analogous
phenomenon is shown for the heat semigroup, as expected.
