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STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMST hroughout, we let [a,b] be a bounded interval ([a,b]) denotesthe space of functions with derivatives of second order continuous up tothe ([a,b]) is the subspace of functions that vanish near a second order differential operator of the formLu(x) =r(x)u (x) +r (x)u (x) +q(x)u(x)=ddx(r(x)dudx)+q(x)u(x).(1)We assume thatr C1([a,b]) andq C0([a,b]) are real, and thatr(x) cfor somec > operatorLis the most general second order real ODE which isformallyself-adjointonL2(dx), in that ba(Lu)v dx= bau(Lv)dx u,v C2c([a,b]).The conditionu,v C2c([a,b]) implies that when integrating by parts theboundary terms vanish. SinceLhas real coefficients, conjugatingvor notdoes not affect the generalu,v C2([a,b]),(2) ba(Lu)v u(Lv)dx=r(u v uv ) baand we need to impose first order conditions onu,vat the endpoints tomake the right hand side conditionBis an expression of the formBu= u(a) + u(b) + u (a) + u (b)for real constants.

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [a;b] be a bounded interval in R. C2([a;b]) denotes the space of functions with derivatives of second order continuous up to

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