Transcription of 6 Sturm-Liouville Eigenvalue Problems
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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.
a few key theorems, though this will not be an extensive review of Sturm-Liouville theory. The interested reader can review the literature and more advanced texts for a more in depth analysis. We define the Sturm-Liouville operator as L = d dx p(x) d dx +q(x). (6.2) The Sturm-Liouville eigenvalue problem is given by the differential equation ...
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