# THE INVARIANCE PRINCIPLE AND ASYMPTOTIC COMPLETENESS FOR A QUANTUM MECHANICAL SYSTEM

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## Abstract

We study operators H and H(,0) acting on the Hilbert Space L('2)(R). The free or unperturbed operator H(,0) is the multiplication (also known as the position) operator x. We consider H = H(,0) + A, where the perturbation A is an integral operator which may be unbounded. H(,0) is known to be self-adjoint. We give conditions on the perturbation A for H to be self-adjoint as well.;The primary objective of this dissertation is to prove the main conclusions of scattering theory for operators of the type just described. Both time-dependent and stationary methods are utilized. We use a factored perturbation technique to prove existence and completeness of the wave operators W (+OR-) (H,H(,0)). We also obtain conditions for the existence of the wave operators for a generalized integral perturbation where A is not factorable. This is accomplished by first finding operators L(,0) and B in momentum (Fourier-transform) space which are equivalent to our operators H and H(,0) in configuration space. Then, by showing that the two pairs of Hamiltonians are equivalent, we have in effect proved that all theorems proved for H and H(,0) are true for L (= L(,0) + B) and L(,0), and vice versa. We can then examine the operators L and L(,0) and transfer our results to H and H(,0).