THE SCATTERING THEORY OF THE KLEIN-GORDON EQUATION IN TWO HILBERT SPACES WITH GENERAL AND OSCILLATING POTENTIALS

Abstract

This dissertation considers three problems associated with the Klein-Gordon Equation: (a) The conditions for the operator to be self-adjoint; (b) The existence of the wave operator; and (c) The completeness of the wave operator. These problems are considered for the operator with general and oscillating potentials.;For problem (a) the work is based on the theory of forms extensions originated by K. Friederichs; and for problems (b) and (c), the abstract theory of scattering which originated in the work of Kato and Birman. The particular result which we use for problems (a) and (b) is the recent theorem proven by M. Schechter, in which he was able to relax requirements on J (no requirement for the bijectivity of J, and no reference to R(z), for example).;Application of the methods described above to the Klein-Gordon operator allowed us to solve the three problems above for an unbounded operator J and also for the oscillating potential.