Continuum quantum systems as limits of discrete quantum systems. II: State functions (Q2735752)

This is the second of a series of four papers by the author, two of which have already appeared [Part I, J. Funct. Anal. 186, No. 1, 153-166 (2001; Zbl 0989.81053); Part III, J. Math. Phys. 42, No. 10, 4653-4668 (2001; Zbl 1018.81030)], devoted to the establishment of a theory of correspondences between discrete and continuum quantum systems, preserving the algebraic structure. This second paper considers the notion of pointwise convergence of vectors, applicable in the presence of function-space representations of the state spaces. Is is shown that it implies the algebraic criterion for convergence, relying on the notion of inductive resolution, that was introduced in the first paper (loc. cit.). As examples the normal approximation theorem and the convergence of the Kravchuk functions to the Hermite-Gaussians are proved to hold also in terms of the algebraic notion of convergence.