Scattering Theory for $N$-Particle Systems with Stark Effect: Asymptotic Completeness

Abstract

The present paper is a continuation to the work [15] where we have proved the non-existence of bound states and the principle of limiting absorption for /V-particle Stark Hamiltonians. We here study the problem of asymptotic completeness of wave operators. The asymptotic completeness for Nparticle quantum systems without uniform electric fields was first proved by Sigal-Soffer [12] and after that remarkable work, alternative proofs have also been given by several authors [2, 7, 13, 17]. We use the local commutator method initiated by Mourre [9] to prove the asymptotic completeness for Nparticle systems with uniform electric fields. The proof is, in principle, based on the same idea as developed in the above works [2, 7, 12, 13, 17] for the case without electric fields. We analyse the propagation properties in the configuration space for solutions to the Schrodinger equation. But the phase space analysis is not required, because charged particles are scattered along only one direction (direction of a given uniform electric field). We shall formulate the problem precisely, fixing several basic notations employed in many-particle scattering theory. We consider a system of /Yparticles moving in a constant electric field £e/2, <?^0. We denote by in.,, e-, and r^e/2, 1^/^A/, the mass, charge and position vector of the /-th particle, respectively. Then, for such a system, the total energy Hamiltonian takes the following form: