Convergence of the Optimized Delta Expansion for the Connected Vacuum Amplitude: Zero Dimensions

TL;DR

This paper proves the convergence of the linear delta expansion for the connected vacuum function in zero-dimensional models, showing an exponential decay of the approximation error with respect to the number of terms.

Contribution

It extends previous convergence results from the vacuum generating functional to the connected vacuum function in zero dimensions, with a specific error rate.

Findings

Convergence of the sequence $W_N$ to $W$ with error proportional to $e^{-c\,\sqrt{N}}$

The convergence rate is influenced by the zeros of $Z_N$

Optimal parameter choice ensures exponential convergence

Abstract

Recent proofs of the convergence of the linear delta expansion in zero and in one dimensions have been limited to the analogue of the vacuum generating functional in field theory. In zero dimensions it was shown that with an appropriate, $N$-dependent, choice of an optimizing parameter $\l$, which is an important feature of the method, the sequence of approximants $Z_N$ tends to $Z$ with an error proportional to ${\rm e}^{-cN}$. In the present paper we establish the convergence of the linear delta expansion for the connected vacuum function $W=\ln Z$. We show that with the same choice of $\l$ the corresponding sequence $W_N$ tends to $W$ with an error proportional to ${\rm e}^{-c\sqrt N}$. The rate of convergence of the latter sequence is governed by the positions of the zeros of $Z_N$.