Born approximation in linear-time invariant system

TL;DR

This paper establishes a mathematical link between linear-time invariant systems and the Helmholtz equation, applying the Born approximation from quantum mechanics to derive solutions as infinite series, enhancing analytical methods for LTI oscillation problems.

Contribution

It introduces a novel application of the Born approximation to LTI systems, connecting them to quantum scattering theory and deriving solution series through this analogy.

Findings

Derived Born series solutions for harmonic forced oscillations.

Established conditions for applying the Born approximation to LTI systems.

Linked LTI oscillation solutions to Feynman graph representations.

Abstract

Linear-time invariant (LTI) oscillation systems such as forced mechanical vibration, series RLC and parallel RLC circuits can be solved by using simplest initial conditions or employing of Green's function of which knowledge of initial condition of the force term is needed. Here we show a mathematical connection of the LTI system and the Helmholtz equation form of the time-independent Schr\"{o}dinger equation in quantum mechanical scattering problem. We apply Born approximation in quantum mechanics to obtain LTI general solution in form of infinite Born series which can be expressed as a series of one-dimensional Feynman graphs. Conditions corresponding to the approximation are given for the case of harmonic driving force. The Born series of the harmonic forced oscillation case are derived by directly applying the approximation to the LTI system or by transforming the LTI system to Helmholtz equation prior to doing the approximation.