The functional integral with unconditional Wiener measure for anharmonic oscillator

TL;DR

This paper introduces an alternative method for calculating the Wiener measure functional integral with a quartic term, using series expansion and parabolic cylinder functions, providing a new approach to anharmonic oscillator problems.

Contribution

It develops a novel method for evaluating the anharmonic oscillator's functional integral via series expansion and special functions, differing from traditional perturbation techniques.

Findings

Series expansions are uniformly convergent.

Recurrence relations for N-dimensional approximation are derived.

Differential equations yield the functional integral in the continuum limit.

Abstract

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).