Remarks on the Feynman representation
There is a substantial literature on Feynman's path integral in quantum physics, but there is still not a satisfactory mathematical theory of integration with respect to such object - unbounded set functions, in the sense that approaches which yield a rich structure do not admit, or have not been proved to admit, sufficiently many integrable functions to provide a representation of solutions to Schrödinger's equation for natural potentials. The present paper deals with an approach to path integrals developed in an earlier paper of the author [J. Funct. Anal. 66, 381-405 (1986; Zbl 0589.46034)]. Not much is known about the class of integrable functions in this sense, except that the integrand of Feynman's representation of the solution of Schrödinger's equation is integrable for a wide class of potentials (ibid.). Proceeding along similar lines, this paper shows that the class of integrable functions is complete for a slightly stronger integration process, which still enables the treatment of Coulomb potentials, but not attractive inverse square potentials. Although completeness is a most desirable property possessed by a locally convex space, it is no easy task to verify that a net converges in the underlying topology of the space of integrable functions. The proof that Coulomb potentials give rise to integrable functions on the space of paths appeals to an extension of the Trotter product formula to the situation of uniform convergence as a parameter varies of a set.
the class of integrable functions is complete
scientific article; zbMATH DE number 3994365