We show that in the absence of a magnetic field the spectrum of the magnetic Schrödinger 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 Schrödinger operator, and hence shows that even in the absence of a magnetic field the magnetic potential can change the asymptotics of the Schrödinger spectrum; that is, the Aharonov–Bohm effect takes place. We also study the Aharonov–Bohm effect for the magnetic Schrödinger operator on a torus.
"The Aharonov–Bohm effect in spectral asymptotics of the magnetic Schrödinger operator." Anal. PDE 7 (1) 245 - 266, 2014. https://doi.org/10.2140/apde.2014.7.245