Scattering and a Plancherel formula of spherical varieties for real reductive split groups

Extended version, 85 pages. Repair of errors from typos to more serious. Main changes in section 8.1 to 8.5We establish the analog for real spherical varieties of the Scattering Theorem of Sakellaridis and Venkatesh (\cite{SV}, Theorem 7.3.1) for p-adic spherical varieties. We use properties of the Harish-Chandra homomorphism of Knop for invariant differential operators of the variety, special coverings of the variety and spectral projections