Application of functional analysis to existence and completeness problems in one-dimensional quantum mechanics

(I) Certain theorems on the existence and completeness of wave operators for one-dimensional quantum systems are stated and proved. (II) Some results concerning alternatives to Feynman's integral over paths formulation of quantum mechanics are obtained. The solution to the one-dimensional Schroedinger equations is formally expressed as a one-parameter group of operators acting on the initial state with the time t the parameter. With t fixed, the corresponding operator is represented as a limit of a sequence, of operators by the Trotter product formula. A simplifying approximation is introduced in this operator sequence and an attempt is made to tie the limit of the approximate sequence to some function space integral. It is found, however, that the measure proposed for finite-dimensional cylinder sets in function space (analogous to the corresponding measure in the Feynman path integral formulation) fails to satisfy a certain consistency condition, and thus cannot be extended to the sigma-field in function space generated by all finite-dimensional measurable cylinder sets. Thus, the approximation does not permit and to express the solution to the one-dimensional Schroedinger equation as a Feynman-type function space integral