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Transactions of the Moscow Mathematical Society

This journal, a translation of Trudy Moskovskogo Matematicheskogo Obshchestva, contains the results of original research in pure mathematics.

ISSN 1547-738X (online) ISSN 0077-1554 (print)

The 2020 MCQ for Transactions of the Moscow Mathematical Society is 0.74.

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A functional integral with respect to a countably additive measure representing a solution of the Dirac equation
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by N. N. Shamarov
Translated by: J. Wiegold
Trans. Moscow Math. Soc. 2005, 243-255
DOI: https://doi.org/10.1090/S0077-1554-05-00150-0
Published electronically: November 16, 2005

Abstract:

A family $\{M_{p}\}_{p\in \mathbb {R}^d}$ of cylindrical measures is constructed on the space of functions ( $=$ trajectories) $f: [0,+\infty )\to \mathbb {R}^d$ such that for every $t\geq 0$ the formula \begin{equation} \psi (t,p)= \int M_{p}(df) \psi _0(f( t)) \end{equation} represents a solution of the Cauchy problem \begin{equation} \frac \partial {\partial t} \psi = i \widehat H \psi \quad (t\geq 0),\quad \psi (0,\cdot )=\psi _0 \end{equation} (with respect to the required function $\psi : [0,\infty )\times \mathbb {R}^d\to \mathbb {C}^s$, ${d,s\in \mathbb {N}}$), 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 $t$, and are Fourier transforms of countably additive (and in general matrix-valued) measures in the space variables.

The images of the measures $M_p$ obtained by restricting trajectories to finite intervals $[0,T]$ 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 (matrix-valued) transition measures that give cylindrical measures $M_p$ similarly to the way Markov transition probabilities give the distribution of a Markov process.

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Bibliographic Information
  • N. N. Shamarov
  • Affiliation: Mechanics and Mathematics Department, Moscow State University, Vorob’evy Gory, Moscow 119234, Russia
  • Published electronically: November 16, 2005
  • Additional Notes: This work was supported by the Russian Foundation for Fundamental Research, Grant No. 02-01-01074.
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Moscow Math. Soc. 2005, 243-255
  • MSC (2000): Primary 28B99; Secondary 46G10, 81Q05
  • DOI: https://doi.org/10.1090/S0077-1554-05-00150-0
  • MathSciNet review: 2193434