Convergence of eigenmodes of a Sturm Liouville operator. Is there any "eassy to see" proof for:
"The eigenmodes of a Sturm Liouville ODE in a closed  interval [a,b], with given boundary conditions, form a complete, orthogonal basis for continuous functions defined in [a,b] meeting the same boundary conditions."???
I found some references, but they seem to be somewhat obscure. I've been trying to get my own proof but I keep hitting the wall.
Thanks.
 A: Actually, there is no need to restrict to functions satisfying the same endpoint conditions. The orthogonal functions always form a complete orthogonal set. Of course, if all the orthonormal functions $\{ e_{n}\}$ satisfy $e_{n}(a)=0$, then you won't get pointwise convergence for $f=\sum_{n}(f,e_{n})e_{n}$ at $a$, but you can get it everywhere else on $(a,b)$ if the function is, for example, differentiable. Regardless, you'll always get convergence in $L^{2}_{w}[a,b]$ with respect to the proper weight $w$, and the eigenfunctions will form a complete orthonormal basis of $L^{2}_{w}[a,b]$. Moreover, for such regular problems $Ly=-(py')y'+qy$, it is possible (under reasonable conditions on the coefficients) to change variables to get an equivalent problem $L_{new}y=-y''+Qy$. In this transformed setting, one can show that the convergence of expansions is equivalent to the convergence of classical expansions where $Q=0$ (keeping the same endpoint conditions, of course.) This equivalence is strong: the Fourier series with $Q \ne 0$ converges at some point $x$ iff the series for the problem where $Q=0$ converges at the same point and, when either converges at $x$, they both converge to the same value. It turns out that the eigenfunctions $y''+Qy=\lambda y$ are asymptotically like those of $y''=\lambda y$ for large $|\lambda|$, and this approximation is very good, so good that the eigenvalues of the general problem can be shown to be asymptotic to those where $Q=0$ as the eigenvalues tend to $\infty$. This is why special functions on non-singular intervals always have familiar asymptotics, too.
