New formulae for the Aharonov-Bohm wave operators

In some recent works on scattering theory [KR1, KR2, KR3, KR5], it was conjectured and then proved that, modulo a compact term, the wave operators for Schrödinger systems can be rewritten as a product of a function of the dilation operator A and a function of the Laplacian −∆. Furthermore, the functions of the dilation operator are rather insensitive to a particular choice of the perturbed operator and depend mainly on the free system and on the space dimension. In this paper, we obtain a similar result for the five-parameter family of Hamiltonians describing the non-relativistic Aharonov-Bohm systems [AT, DS]. More precisely, we first show that the wave operators for the original Aharonov-Bohm Hamiltonian [AB, R] can be rewritten as explicit functions of A only. For the wave operators corresponding to other self-adjoint extensions, we prove that the additional terms are given by the product of a function of A and a function of −∆. Let us already stress that the functions of the dilation operator depend on the flux of the magnetic field, but not on the other parameters of the boundary condition at 0 ∈ R2. These new formulae might serve for various further investigations on scattering theory for systems with less singular magnetic fields. In particular, it would interesting to study the structure of the wave operators for Schrödinger operators with magnetic fields supported in small sets, see for example [EIO, T1, T2]. These new expressions also lead to a topological approach of Levinson’s theorem. In this respect, we mention two papers related to Levinson’s for the original Aharonov-Bohm operator [L, SM]. We intend to address both subjects in forthcoming publications. The structure of this paper is the following: We first recall the constructions of the fiveparameter family of self-adjoint operators, mainly borrowed from [AT]. After a technical interlude on the Fourier transform and on the generator of dilations, we show in Theorem 4 that