Operator-valued Feynman integrals of certain finite-dimensional functionals
Authors:
G. W. Johnson and D. L. Skoug
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
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Abstract | References | Similar Articles | Additional Information
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.
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- R. H. Cameron and D. A. Storvick, An operator valued function space integral and a related integral equation, J. Math. Mech. 18 (1968), 517–552. MR 0236347, DOI https://doi.org/10.1512/iumj.1969.18.18041
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- D. L. Skoug, Generalized Ilstow and Feynman integrals, Pacific J. Math. 26 (1968), 171–192. MR 230870
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Additional Information
Keywords:
Feynman integral,
Wiener integral,
convolution operator,
Fourier transform,
operator valued Feynman integral
Article copyright:
© Copyright 1970
American Mathematical Society