Scattering Problem for Local Perturbations\newline of the Free Quantum Gas

Abstract:

Scattering theory for perturbations of the intrinsic Dirichlet (Laplace–Beltrami) operator H ₀=−div^Γ∇^Γ on L ₂(Γ,π _z ), i. e. the space of π _z -square integrable functions on the configuration space Γ over ℝ^d, is studied. Here π _z denotes Poisson measure with intensity z. We show that for an arbitrary regular non-zero potential V the standard wave operators W ^±(H ₀,H ₀+V) do not exist, and propose to consider Dirichlet operators of perturbed Poisson measures instead of potential perturbations of the Hamiltonian H ₀. As case studies, cylindric smooth densities and finite volume Gibbs perturbations of the Poisson measure are considered. In these cases the existence of the corresponding wave operators is proved.