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Chapter 5 Sturm-Liouville Theory - Texas Tech University

Chapter 5 Sturm-Liouville Oscillation and Separation TheoryConsider the di erential equationa2(x)y00+a1(x)y0+a0(x)y= 0( )wherea2(x) is not zero for allx2[a;b],ai(x)2C[a;b]. Rewrite ( ) in the formy00+a1a2y0+a0a2y=y00+p(x)y0+q(x)y=0D e nek(x)=eRp(s)ds;Thenddx(k(x)y0)+k(x)q(x) y=0or(ky0)0+g(x)y= 0( )De ne the di erential operatorsL(y)=(ky0)0+g(x)yM(y)=a2y00+a1y 0+a0y( )The adjoint of M is de ned byM(y)=(a2y)00 (a1y)0+a0y12 Chapter 5. Sturm-Liouville THEORYM(y)=a2y00+(2a02 a1)y0+(a002 a01+a0)y:After some manipulation it is easy to show thatvM(u) uM(v)=[(a1 a02)vu+a2(vu0 uv0)]0:This result is calledLaGrange's identityand we rewrite it asvM(u) uM(v)=ddxP(u;v):By an integration we obtainGreen's formula,Zba vM(u) uM(v) dx=P(u;v)jbaIfM(u)=M(u), the equationM(u) = 0 is said to be self-adjoint. HenceM(u)=0isself-adjoint ifa02=a1:In this case Lagrange's identity becomesvM(u) uM(v)=[a2(vu0 uv0)]0=[a2(x)N(v;u)]0andM(u)=(a2u0)0+a0u Clearly the operatorLde ned by ( ) is self-adjoint and the discussion preceding ( )shows every general linear equation can be put into self adjoint Separation theoremsTheorem [ sturm Separation Theorem]1.

Roughly speaking, the Sturm Separation theorem states that linearly independent solu- tions have the same number of zeros. If we consider two difierent equations, for example

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