High-velocity estimates for Schrödinger operators in two dimensions: Long-range magnetic potentials and time-dependent inverse scattering

Abstract

We introduce a general class of long-range magnetic potentials and derive high velocity limits for the corresponding scattering operators in quantum mechanics, in the case of two dimensions. We analyze the high velocity limits that we obtain in the presence of an obstacle and we uniquely reconstruct from them the electric potential and the magnetic field outside the obstacle, that are accessible to the particles. We additionally reconstruct the inaccessible fluxes (magnetic fluxes produced by fields inside the obstacle) modulo 2π, which give a proof of the Aharonov–Bohm effect. For every magnetic potential A in our class, we prove that its behavior at infinity [Formula: see text] can be characterized in a natural way; we call it the long-range part of the magnetic potential. Under very general assumptions, we prove that [Formula: see text] can be uniquely reconstructed for every [Formula: see text]. We characterize properties of the support of the magnetic field outside the obstacle that permit us to uniquely reconstruct [Formula: see text] either for all [Formula: see text] or for [Formula: see text] in a subset of 𝕊1. We also give a wide class of magnetic fields outside the obstacle allowing us to uniquely reconstruct the total magnetic flux (and [Formula: see text] for all [Formula: see text]). This is relevant because, as it is well-known, in general the scattering operator (even if it is known for all velocities or energies) does not define uniquely the total magnetic flux (and [Formula: see text]). We analyze additionally injectivity (i.e. uniqueness without giving a method for reconstruction) of the high velocity limits of the scattering operator with respect to [Formula: see text]. Assuming that the magnetic field outside the obstacle is not identically zero, we provide a class of magnetic potentials for which injectivity is valid.