Transcription of Sturm-Liouville Theory - Oregon State University
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Sturm-Liouville Theory - Oregon State University
Sturm-Liouville Theory (boundary value)A second order sturm liouville problem is a homoge-neous boundary value problem of the form[P(x)y ] +Q(x)y+ w(x)y= 0 1y(a) + 1y (a) = 0 2y(b) + 2y (b) = 0whereP,P ,Q,ware continuous and real on [a,b], andPandware : Fory1andy2two linearly independent solu-tions of the homogeneous differential equation,nontriv-ialsolutions of the homogeneous boundary value problemexist iffdet 1y1(a) + 1y 1(a) 1y2(a) + 1y 2(a) 2y1(b) + 2y 1(b) 2y2(b) + 2y 2(b) = 0 Definition: Values of for which nontrivial solutions existare called eigenvalues. The corresponding solutions arecalled : The eigenvalues of a homogeneous sturm -Liouvilleproblem are real and non-negative and can be arrangedin a strictly increasing infinite sequence0 1< 2< 3< ..and n asn .Theorem: For each eigenvalue, there exists exactly onelinearly independent eigenfunction,yn.
Theorem: The eigenvalues of a homogeneous Sturm-Liouville problem are real and non-negative and can be arranged in a strictly increasing in nite sequence
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