The notion of a non-central time operator is investigated in the context of an n-dimensional complex Hilbert space, by first abstracting from the canonical commutation relations expressed in the Schroedinger realization when the momentum operator is expressed as a derivative with respect to the coordinate operator spectrum. In a finite dimensional context, the concept of a "spectral derivative" with respect to an operator is developed. The spectral derivative is first defined to operate on the vectors of the Hilbert space, and then extended to operate on the C*-algebra of linear operators acting on the Hilbert space. The complications of the limit as the dimensional becomes infinite is briefly discussed. The procedures for solving spectral-differential eqations are also investigated. The necessary appearance of cyclic operators and discrete Fourier transforms is notable.

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Created: 1997
Last Updated: May 28, 2000
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