6 Sturm-Liouville Eigenvalue Problems

6 Sturm-Liouville Eigenvalue IntroductionIn the last chapters we have explored the solution of boundary value problemsthat led to trigonometric eigenfunctions. Such functions can be used to repre-sent functions in Fourier series expansions. We would like to generalize someof those techniques in order to solve other boundary value Problems . A class ofproblems to which our previous examples belong and which have eigenfunc-tions with similar properties are the Sturm-Liouville Eigenvalue Problems involve self-adjoint (differential) operators which play an im-portant role in the spectral theory of linear operators and the existence of theeigenfunctions we described in Section These ideas will be introducedin this physics many Problems arise in the form of boundary value problemsinvolving second order ordinary differential equations. For example, we mightwant to solve the equationa2(x)y +a1(x)y +a0(x)y=f(x)( )subject to boundary conditions.

Liouville form: d dx p(x) dy dx +q(x)y = F(x). (6.5) Another way to phrase this is provided in the theorem: Theorem 6.1. Any second order linear operator can be put into the form of the Sturm-Liouville operator (6.2). The proof of this is straight forward, as we shall soon show. Consider the equation (6.1). If a …

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