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

The Aharonov-Bohm effect in spectral asymptotics of the magnetic Schr\"odinger operator