Eigenvalue Problems

Abstract

In obtaining Legendre polynomials as solutions to Legendre’s equation,

$$(1\, - \,x^2 )\,P''(x)\, - \,2xP'(x)\, + \,\lambda P(x)\, = \,0,$$

we have solved a typical eigenvalue problem. Given an equation (or set of equations) containing a parameter (here l), we seek solutions that satisfy some special requirement (e.g., the series must converge for x = ±1). To obtain such solutions, we must choose particular values(eigenvalues) for the parameter. In this case,

$$\lambda \, = \,n(n + 1)\,\,\,\,\text{with}\,\,\,\,n\, = \,0,\,1,\,2, \ldots$$

That is, polynomial solutions (which are required for convergence atx = ±1) arise only for certain values of λ and not for all values.