Convergence proof for optimized<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>δ</mml:mi></mml:math>expansion: Anharmonic oscillator

Abstract

A recent proof of the convergence of the optimized $\ensuremath{\delta}$ expansion for one-dimensional non-Gaussian integrals is extended to the finite-temperature partition function of the quantum anharmonic oscillator. The convergence is exponentially fast, with the remainder falling as ${e}^{\ensuremath{-}c{N}^{\frac{2}{3}}}$ at order $N$ in the expansion, independently of the size of the coupling or the sign of the mass term. In particular, the approach gives a convergent resummation procedure for the double-well (non-Borel-summable) case.