PROOF OF THE CONVERGENCE OF THE LINEAR DELTA-EXPANSION - ZERO DIMENSIONS

The convergence of the linear delta expansion is studied in the context of the integral I: integral-infinity/-infinity e(-gx4)dx, which corresponds to massless phi4 theory in 0 dimensions. The method consists of rewriting the exponent as -deltagx4-lambda(1-delta)x2 and expanding in powers of delta. The arbitrary parameter lambda is fixed by the principle of minimal sensitivity, partial derivative I(K)(lambda)/partial derivative lambda = 0, where I(K) is the expansion truncated at order K with delta set equal to 1. This has a solution lambda(K)BAR only for K odd, when it gives very good numerical results. We are able to show analytically, using saddle-point methods, that the sequence of approximants I(K) (lambda(K)BAR) is convergent, the error decreasing exponentially with K, even though for fixed lambda the series expansion is a divergent alternating series.