Evaluating the Stieltjes transform via a modified density reconstruction basis

Abstract

The Stieltjes transform is a powerful resummation technique for evaluating a divergent alternating perturbation series by mapping the coefficients to the moments of some density function. There are infinitely many basis sets that can be used to reconstruct this density function and the existing scheme utilizes Laguerre basis with weight e^−x/2. In this work, we investigate the sensitivity of the rate of convergence of the Stieltjes transform to the choice of basis by proposing the weight exp(−√x). The corresponding orthonormal polynomials are constructed using the Gram-Schmidt process and Shohat-Favard theorem. By applying the original and modified schemes, we demonstrate the convergence to the exact value regardless of the chosen basis. The modified weight converges more slowly than the original in the non-perturbative regime but faster in the perturbative regime, implying the significant effect of weights in the rate of convergence. Our findings suggest that an optimal basis set exist to access the strong-coupling regime using a minimal number of moments.