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# ch4 - Chapter 4 Sturm-Liouville problems 4.1 Regular...

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Chapter 4: Sturm-Liouville problems 4.1. Regular Sturm-Liouville problems In the previous applications we arrived via separation of variables to some linear homogeneous ODE of second order on an interval ( a, b ), together with some homogeneous boundary conditions. In this chapter we are going to anal- yse a large class of those problems, and we will see that they lead to certain orthogonal bases on R ( a, b ). Assume that [ a, b ] is a bounded interval on IR, and let the inner product be given by < f, g > = b a f ( x ) g ( x ) dx. Suppose L is a second-order linear DOP, defined for functions f C 2 ([ a, b ]), L ( f ) := rf + r f + pf = ( rf ) + pf, (1) where r, p C 2 ([ a, b ]). We shall assume that the leading coefficient r is non- vanishing on [ a, b ], since the existence of ”singular points” where r = 0 con- siderably complicates the theory. The following simple result states that the operator L above is formally self-adjoint (in the terminology of linear algebra). Lemma 4.1. (Lagrange’s identity) If f, g C 2 ([ a, b ]) , then L ( f ) , g = f, L ( g ) + r ( f g - f g ) b a . (2) Proof: Using integration by parts we have b a ( rf ) g dx = [ rf g ] b a - b a rf g dx = rf g - rf g b a + b a f r g dx, and the result follows. Evidently, the discrepancy between formal and actual self-adjointness lies in the endpoint terms in (2). They can be eliminated by restricting the problem 25

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to functions which satisfy suitable boundary conditions. More precisely, it is appropriate to impose two independent boundary conditions of the form B 1 ( f ) = α 1 f ( a ) + α 1 f ( a ) + β 1 f ( b ) + β 1 f ( b ) = 0 , (3) B 2 ( f ) = α 2 f ( a ) + α 2 f ( a ) + β 2 f ( b ) + β 2 f ( b ) = 0 , (4) where the α ’s and β ’s are constants. We say that the boundary operators B 1 , B 2 in (3),(4) are self-adjoint (relative to the operator L ), if r ( f g - f g ) b a = 0 for all f, g satisfying (3),(4). Most boundary conditions that arise in practice are separated , meaning that each one of (3),(4) involves a condition at only one endpoint, αf ( a ) + α f ( a ) = 0 , βf ( b ) + β f ( b ) = 0 , (5) where ( α, α ) = (0 , 0), and ( β, β ) = (0 , 0). It is easy to check that separated boundary conditions are always self-adjoint (relative to any operator L ). There is also one set of nonseparated boundary conditions that is commonly used, the periodic boundary conditions f ( a ) = f ( b ) , f ( a ) = f ( b ) . (6) These are self-adjoint relative to L provided that r ( a ) = r ( b ) . We are now ready to formulate the boundary value problems that lead to orthogonal bases for R ( a, b ). Definition 4.2. Let a formally self-adjoint DOP L and two boundary operators B 1 , B 2 , which are self-adjoint for L , be given by (1) and (3),(4), respectively, where r, r and p are real and continuous on [ a, b ] and r > 0 on [ a, b ] , and let w be a positive, continuous function on [ a, b ] . 26
The object is to find all solutions ( f, λ ) , ( f C 2 ([ a, b ]) , λ IR), of the boundary value problem ( StL ) L ( f ) + λwf = 0 , B 1 ( f ) = B 2 ( f ) = 0 . (7) The problem (StL) is called a regular Sturm Liouville problem.

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ch4 - Chapter 4 Sturm-Liouville problems 4.1 Regular...

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