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Define the Sturm-Liouville operator $L$ on $[a,b]$ such that $Lu = \frac{1}{\omega}[-(pu')'+qu]$, $p$ and $\omega$ being strictly positive real valued function, and $q$ a positive real valued function.

The inner product is defined as such : $\int_a^b u(x)\overline{v(x)}\omega(x) \, \mathrm{d}x $

The question asks to prove that the operator is positive, i.e. $(Lu,u)\ge0$ (I believe they really mean positive semi-definite).

I got so far:

$(Lu,u)= \int_a^b q|u|^2 \mathrm{d}x + \int_a^b p|u'|^2 \mathrm{d}x - [pu'\bar{u}]^b_a $

Now I don't really understand how the boundary conditions come into play... any one knows?

nayriz
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  • I can't answer your question fully and don't know about your knowledge but would looking at orthogonal polynomials help? They are normally solution of a very similar kind of ODE to that of the Sturm-Liouville oprator.

    Sorry if this is something you already know but since you don't mention perhaps it could be worth a look.

     – user88595 Mar 01 '14 at 22:56
  • Here is a different question with a closely related problem. – ccorn Mar 02 '14 at 02:26

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