Solution to the Equations of the Moment Expansions

TL;DR

This paper introduces a new formula for matching Taylor and asymptotic expansions, tested on quantum Hamiltonian moments, improving the understanding and application of moment expansions in quantum mechanics.

Contribution

It develops a novel matching formula for Taylor and exponential asymptotic expansions, applied to quantum Hamiltonian moments, revealing new insights into the connected-moments expansion.

Findings

Accurately computes energies and overlaps for quantum systems.

Reveals overlooked features of the connected-moments expansion.

Demonstrates effectiveness on harmonic and anharmonic oscillators.

Abstract

We develop a formula for matching a Taylor series about the origin and an asymptotic exponential expansion for large values of the coordinate. We test it on the expansion of the generating functions for the moments and connected moments of the Hamiltonian operator. In the former case the formula produces the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We choose the harmonic oscillator and a strongly anharmonic oscillator as illustrative examples for numerical test. Our results reveal some features of the connected-moments expansion that were overlooked in earlier studies and applications of the approach.