Combining Continuous and Discrete Phenomena for Feynman’s Operational Calculus in the Presence of a (C₀) Semigroup and Feynman–Kac Formulas with Lebesgue–Stieltjes Measures

This paper introduces the presence of a (C₀) semigroup of linear operators into the disentangling of functions of noncommuting operators in the setting where time-ordering measures with finitely supported discrete parts are allowed. Some examples are discussed. Furthermore, in this more general setting the main result of this paper, an evolution equation satisfied by disentangled exponential functions is obtained. A related generalized integral equation is also discussed. Via some detailed examples, Feynman–Kac formulas (Volterra–Stieltjes integral equations) with Lebesgue–Stieltjes measures are also derived and an associated differential equation is derived.