Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 8Factorization Method

Factorization method is an elegant way to solve Sturm–Liouville systems. It basically allows us to replace a Sturm–Liouville equation, a second-order linear differential equation, with a pair of first-order differential equations. For a large class of problems the method immediately yields the eigenvalues and allows us to write the ladder operators for the problem. These operators are then used to construct the eigenfunctions from a base function. Once the base function is normalized, the manufactured eigenfunctions are also normalized and satisfy the same boundary conditions as the base function. We first introduce the method of factorization and its basic features in terms of five theorems. Next, we show how the eigenvalues and the eigenfunctions are obtained and introduce six basic types of factorization. In fact, factorization of a given second-order differential equation is reduced to identifying the type it belongs to. To demonstrate the usage of the method, we discuss the associated Legendre equation and the spherical harmonics in detail. We also discuss the radial part of the Schrödinger equation for the hydrogen-like atoms. The Gegenbauer polynomials, symmetric top, Bessel functions, and the harmonic oscillator problem are the other examples we discuss in terms of the factorization method. Further details and an extensive table of differential equations that can be solved by this technique is given by Infeld and Hull [[3]], where this method ...