The functional integral with unconditional Wiener measure for anharmonic oscillator

In this article we propose the calculation of the unconditional Wiener measure functional integral with a term of the fourth order in the exponent by an alternative method as in the conventional perturbative approach. In contrast to the conventional perturbation theory, we expand into power series t...

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Bibliographic Details
Published in:Journal of mathematical physics 2008-11, Vol.49 (11), p.113505-113505-27
Main Authors: BOHDACIK, J, PRESNAJDER, P
Format: Article
Language:English
Subjects:
ANHARMONIC OSCILLATORS
Approximation
APPROXIMATIONS
Calculus
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
CONVERGENCE
DIFFERENTIAL EQUATIONS
Exact sciences and technology
FUNCTIONAL ANALYSIS
HARMONIC OSCILLATORS
INTEGRAL EQUATIONS
INTEGRALS
Mathematical methods in physics
MATHEMATICAL SOLUTIONS
Mathematics
MEASURE THEORY
PERTURBATION THEORY
Physics
POWER SERIES
RECURSION RELATIONS
Sciences and techniques of general use
STOCHASTIC PROCESSES
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Summary:In this article we propose the calculation of the unconditional Wiener measure functional integral with a term of the fourth order in the exponent by an alternative method as in the conventional perturbative approach. In contrast to the conventional perturbation theory, we expand into power series the term linear in the integration variable in the exponent. In such a case we can profit from the representation of the integral in question by the parabolic cylinder functions. We show that in such a case the series expansions are uniformly convergent and we find recurrence relations for the Wiener functional integral in the N -dimensional approximation. In continuum limit we find that the generalized Gelfand–Yaglom differential equation with solution yields the desired functional integral (similarly as the standard Gelfand–Yaglom differential equation yields the functional integral for linear harmonic oscillator).
ISSN:0022-2488
1089-7658
DOI:10.1063/1.3013803