Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Operator-valued Feynman integrals of certain finite-dimensional functionals
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by G. W. Johnson and D. L. Skoug
Proc. Amer. Math. Soc. 24 (1970), 774-780
DOI: https://doi.org/10.1090/S0002-9939-1970-0254675-1

Abstract:

Let ${C_0}[a,b]$ denote the space of continuous functions $x$ on $[a,b]$ such that $x(a) = 0$. Let $F(x) = {f_1}(x({t_1})) \cdots {f_n}(x({t_n}))$ where $a = {t_0} < {t_1} < \cdots < {t_n} = b$. Recently, Cameron and Storvick defined an operator-valued “Feynman Integral.” In their setting, we give a strong existence theorem as well as an explicit formula for the “Feynman Integral” of functionals $F$ as above under weak restrictions on the ${f_i}$’s. We also give necessary and sufficient conditions for the operator to be invertible and an explicit formula for the inverse.
References
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 24 (1970), 774-780
  • MSC: Primary 47.70; Secondary 28.00
  • DOI: https://doi.org/10.1090/S0002-9939-1970-0254675-1
  • MathSciNet review: 0254675