Limiting Absorption Principles and Wave-Operators on L1(μ) Spaces: Applications to Transport Theory

Let X = L1(μ) and let {S(t); t ∈ R} be a positive and bounded (c0)-group of linear operators on X with generator T. Let B ∈ L(D(T); X) be a positive operator and let V(t); t ∈ O} be the (c0)-group with generator T + B. We prove that the boundedness of {V(t); t ≤ 0}, {V(t); t ≤ 0} as well as the existence of the wave operators limt → ±∞V(t) S(−t), limt → +∞S(−t) V(t) are intimately connected to the existence of the strong limits limλ →0±B(λ− T)−1 and to their spectral radii. An optimal theory is given. Applications to scattering theory for transport operators are also given.