Rigorous Operator-Theoretic Generalization of Wave Function Dynamics via Extended Gelfand Triples and Non-Commutative Spectral Deformations

This preprint presents a rigorous operator-theoretic framework for wave function dynamics, extending the conventional Hilbert space formulation via extended Gelfand triples (rigged Hilbert spaces). We investigate non-commutative spectral deformations and their implications for generalized quantum states. The approach provides a robust mathematical foundation for handling continuous spectra and singular potentials in quantum mechanics, complete with explicit operator constructions, domain characterizations, and spectral properties.