Miguel Ballesteros

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

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 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 can be uniquely reconstructed for every . We characterize properties of the support of the magnetic field outside the obstacle that permit us to uniquely reconstruct either for all or for 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 for all ). 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 ). 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 . Assuming that the magnetic field outside the obstacle is not identically zero, we provide a class of magnetic potentials for which injectivity is valid.

Aharonov-Bohm Effect and High-Momenta Inverse Scattering for the Klein-Gordon Equation

We analyze spin-0 relativistic scattering of charged particles propagating in the exterior,

Quantum Electrodynamics of Atomic Resonances

{\Lambda \subset \mathbb{R}^3}

Existence and construction of resonances for atoms coupled to the quantized radiation field

Λ⊂R3, of a compact obstacle

Analyticity of Resonances and Eigenvalues and Spectral Properties of the massless Spin-Boson Model

{K \subset \mathbb{R}^3}

Indirect retrieval of information and the emergence of facts in quantum mechanics

K⊂R3. The connected components of the obstacle are handlebodies. The particles interact with an electromagnetic field in Λ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov–Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: we give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes’ theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo 2π. We additionally give a simple formula for the high momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo π and the simple expression of the high-momenta limit of the scattering operator does not hold true.