Cyclic Operator Decomposition for Solving the Differential Equations

TL;DR

This paper introduces Cyclic Operator Decomposition, a versatile method for solving linear operator equations that generalizes perturbation theory without requiring small parameters, applicable to various differential equations in physics.

Contribution

The paper proposes a new operator series method for solving linear differential equations, allowing flexible choice of operators and generating functions, extending beyond traditional perturbation techniques.

Findings

Successfully applied to classical oscillator, Schrödinger, and wave equations

Provides analytical and numerical solutions without small parameters

Generalizes perturbation theory for broader applicability

Abstract

We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The particular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the generating function. The method allows us to choose the cyclic operators and corresponding generating function selectively, depending on initial problem for analytical or numerical study. Our approach includes, as a particular case, the perturbation theory, but generally does not require inside any small parameters and unperturbed solutions. We demonstrate the applicability of the method to the analysis of several differential equations in mathematical physics, namely, classical oscillator, Schr\"odinger equation, and wave equation in dispersive medium.