Asymptotic evolution of certain observables and completeness in Coulomb scattering. I

Large time behaviour of someN-body systems

“…Now we are ready to formulate the conditions needed for completeness to follow. We recall that in the two-body case in the methods of Enss [26] or Sinha and Muthuramalingam [25] they conclude that whenever the scattering states evolve out of bounded regions in space (Local Decay), the asymptotic observables corresponding to the average velocity x/t and the momentum P are the same (in units where the reduced mass of the system is 1). We conclude the same result here.…”

“…Namely, the local decay and the lower energy decay of the total evolution of the scattering states (formulated as the conditions (LD) and (LED) later in the section). We make use of the theory of evolution of observables developed by Sinha and Muthuramalingam [25] and Enss [26]. We denote by F() the spectral projections and by K all absolute constants that occur in estimates.…”

“…Henceforth for any real valued continuous function φ, we set Given this condition the proof of the theorem on asymptotic observables is almost identical to the one for the two-body case [25,26]. Hence we state it below in the form that we need, without proof.…”

New $L^p$ Estimates for Solutions to the Schrödinger Equations and Time Asymptotic Behavior of Observables

“…Concerning scattering solutions, it is known that the expectation of the potential energy (2) and the difference between the momentum of classical mechanics and that of quantum mechanics (3) decay as t->±oo under suitable conditions (see [4] [6] [27]). Time asymptotic behavior of observables under the Schrodinger group has been of importance in proving asymptotic completeness of the wave operators (see [5] [6][27] [33]). …”