  We show that in the absence of a magnetic field the spectrum of the magnetic
Schr\"odinger operator in an annulus depends on the cosine of the flux
associated with the magnetic potential. This result follows from an analysis of
a singularity in the wave trace for this Schr\"odinger operator, and hence
shows that even in the absence of a magnetic field the magnetic potential can
change the asymptotics of the Schr\"odinger spectrum, i.e. the Aharonov-Bohm
effect takes place. We also study the Aharonov-Bohm effect for the magnetic
Schr\"odinger operator on a torus.
