Abstract
A definition is proposed of integrals of the form\(\int\limits_{ - \infty }^\infty {L(\lambda )} dE_\lambda \)f, whereL(λ) is a family of linear operators and {Eλ} is a resolution of the identity. If we assume the wave and scattering operators of time-dependent scattering theory to exist, these operators can be rigorously expressed, for a wide class of scattering systems, in terms of integrals of the above kind, cach integration corresponding to a momentum variable being placed on the energy shell. If we apply the arguments to potential scattering, assuming the potential function to be square-integrable, the familiar expressions for the wave and scattering operators in terms of partial-waveT-matrix elements are rigorously derived.
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