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Unformatted text preview: 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 > = Z 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 00 + 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 h L ( f ) , g i = h f, L ( g ) i + h r ( f g- f g ) i b a . (2) Proof: Using integration by parts we have Z b a ( rf ) g dx = [ rf g ] b a- Z b a rf g dx = h rf g- rf g i b a + Z b a f r g dx, and the result follows. 2 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 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 h r ( f g- f g ) i 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 ) 6 = (0 , 0), and ( β, β ) 6 = (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 > on [ a, b ] , and let w be a positive, continuous function on [ a, b ] ....
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