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A charged quantum particle in a magnetic field is described by the Schrödinger equation with a magnetic potential. In the Aharonov–Bohm case, the magnetic field is concentrated inside a very thin solenoid, but the magnetic flux still influences the quantum motion outside it.

In joint work with Ari Laptev, we study the time-dependent Schrödinger equation for a two-dimensional harmonic oscillator in an Aharonov–Bohm field. We obtain a global semiclassical description of the corresponding propagator away from trajectories passing through the origin. Away from this singular set, the result shows that the evolution has the same kind of global semiclassical description as in the regular harmonic oscillator. I will begin by introducing the basic model and the main tool used to approximate the integral kernel of the propagator, and then turn to the Aharonov–Bohm case.