Combining Continuous and Discrete Phenomena for Feynman’s Operational Calculus in the Presence of a (\({\varvec{C}}_{\mathbf{0 }}\)) Semigroup and Feynman–Kac Formulas with Lebesgue–Stieltjes Measures

Article

Abstract

This paper introduces the presence of a \((C_{0})\) 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.

Keywords

Disentangling Evolution equation Lebesgue–Stieltjes measure Operational calculus Feynman–Kac formulas 

Mathematics Subject Classification

Primary 44A45 28C05 47D06 58D25n Secondary 45D99 46G12 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCreighton UniversityOmahaUSA

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