Nonperturbational "Continued-Fraction" Spin-offs of Quantum Theory's Standard Perturbation Methods

TL;DR

This paper introduces a nonperturbational continued-fraction scheme derived from standard perturbation methods in quantum theory, offering a potentially more robust approach to solving Schrödinger equations.

Contribution

It reveals a novel continued-fraction scheme that arises from perturbation methods, applicable in certain representations and capable of handling unbounded particle numbers.

Findings

The scheme only works when all components of the inhomogeneous term are nonzero.

It provides a new perspective on quantum field perturbations and unperturbed particle number.

Successive iterations involve denominators, avoiding infinities.

Abstract

The inherently homogeneous stationary-state and time-dependent Schroedinger equations are often recast into inhomogeneous form in order to resolve their solution nonuniqueness. The inhomogeneous term can impose an initial condition or, for scattering, the preferred permitted asymptotic behavior. For bound states it provides sufficient focus to exclude all but one of the homogeneous version's solutions. Because of their unique solutions, such inhomogeneous versions of Schroedinger equations have long been the indispensable basis for a solution scheme of successive perturbational corrections which are anchored by their inhomogeneous term. Here it is noted that every such perturbational solution scheme for an inhomogeneous linear vector equation spins off a nonperturbational continued-fraction scheme. Unlike its representation-independent antecedent, the spin-off scheme only works in representations where all components of the equation's inhomogeneous term are nonzero. But that requirement seems to confer theoretical physics robustness heretofore unknown: for quantum fields the order of the perturbation places a bound on unperturbed particle number, the spin-off scheme contrariwise has only basis elements of unbounded unperturbed particle number. It furthermore is difficult to visualize such a continued-fraction spin-off scheme generating infinities, since its successive iterations always go into denominators.