User:John G. Fletcher/Sandbox 1

The account of this former contributor was not re-activated after the server upgrade of March 2022.

Let us adopt Paul's definition of Λ. Then the Sturm-Liouville equation is Λy(x) = λw(x)y(x). (See eq. 1a in Paul's revision of the original Wikipedia article.) The original article multiplies this equation on the left by 1/w(x) and gets

(1/w(x))[Λy(x)] = λy(x) or

Ly(x) = λy(x), where the operator L = (1/w(x))Λ ;

this is of the form of an eigenvalue equation with eigenvalue λ and eigenvector y(x).

The notion of adjoint is dependent on the notion of scalar product. The original article uses the following weighted scalar product of two vectors z'(x) and z(x):

<z',z> = ∫z'(x)*z(x)w(x)dx ,

where * indicates complex conjugate. So

<y',Ly> = ∫y'(x)*[Ly(x)]w(x)dx

= ∫y'(x)*(1/(w(x))[Λy(x)]w(x)dx = ∫y'(x)*Λy(x)dx

= (∫y(x)*Λy'(x)dx)* = <y,Ly'>* ;

that is, L is self-adjoint. (I have not given details of the transition to the last line, because I believe that all parties agree to it.)

Peter notes that there are two scalar products being used, the second being the unweighted product

<z',z> = ∫z'(x)*z(x)dx .

With respect to this scalar product the operator √(1/w(x))Λ√(1/w(x)) is self adjoint, with eigenvalues the same as above but with different eigenvectors, namely √(w(x))y(x). Using this unweighted product Paul obtains essentially the same results as the original Wikipedia article does with the weighted product. Working with unweighted objects usually is a simplification resulting in shorter expositions, but in this case the exposition is longer. For this reason I believe that introducing the unweighted product is an unnecessary complication.

My conclusion is that all parties are mostly correct, except when they claim that the others are not.