A functional integral with respect to a countably additive measure representing a solution of the Dirac equation
Author:
N. N. Shamarov
Translated by:
J. Wiegold
Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom 66 (2005).
Journal:
Trans. Moscow Math. Soc. 2005, 243255
MSC (2000):
Primary 28B99; Secondary 46G10, 81Q05
Published electronically:
November 16, 2005
MathSciNet review:
2193434
Fulltext PDF Free Access
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Abstract: A family of cylindrical measures is constructed on the space of functions (trajectories) such that for every the formula  (1)  represents a solution of the Cauchy problem  (2)  (with respect to the required function , ), for the general evolution equation from some class containing the classical Dirac equation and the Schrödinger equation in its impulse representation, with ``model'' potentials that are independent of , and are Fourier transforms of countably additive (and in general matrixvalued) measures in the space variables. The images of the measures obtained by restricting trajectories to finite intervals have bounded variation and are countably additive. The integral kernels (``Green's functions'') of the corresponding solution operators, which can be approximated (using Trotter's formula) by integrals of finite multiplicity of the expressions explicitly defined by the ingredients of the original equation, are (matrixvalued) transition measures that give cylindrical measures similarly to the way Markov transition probabilities give the distribution of a Markov process.
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Additional Information
N. N. Shamarov
Affiliation:
Mechanics and Mathematics Department, Moscow State University, Vorob’evy Gory, Moscow 119234, Russia
DOI:
http://dx.doi.org/10.1090/S0077155405001500
PII:
S 00771554(05)001500
Published electronically:
November 16, 2005
Additional Notes:
This work was supported by the Russian Foundation for Fundamental Research, Grant No. 020101074.
Article copyright:
© Copyright 2005
American Mathematical Society
