I know that there are many results about the spectra of compact differential operators but I really don't understand the significance of them. Let's consider, for simplicity, the differential operator:
$Ly = p(x)y'' + q(x)y' + r(x)$ where $p(x),q(x),r(x)$ satisfy some regularity conditions.
If I am interested in solving, or understanding the behavior of solutions $y$ to $Ly = 0$ on $x \in [0,1]$ subject to $y(0) = \alpha, y(1) = \beta$, what benefit would I get by solving for the eigenvalues and eigenfunctions of $L$?
In other words, why would solving the associated eigenvalue problem be beneficial if, to begin with, you are not solving an eigenvalue problem nor any problem that looks like $Ly = \lambda y + f$? (For these types of problems I do see an explanation why solving for $Lv = \lambda v$ could be handy.)
Also, can anyone point me to references for solving the eigenvalues/eigenfunctions numerically of non-constant coefficient operators? The examples I've seen in books only deal with constant coefficients.
Suggestions appreciated.
If you want to solve $Ly=0$ subject to $y(0)=\alpha, y(1)=\beta$, then you can reduce this to an inhomogeneous problem with homogeneous conditions by letting $$ y= w-\alpha x-\beta(1-x). $$ Then $0=Ly=Lw-\alpha Lx-\beta L(1-x)$ gives an equivalent equation for $w$: $$ Lw = (\alpha+\beta)Lx+\beta L1,\;\; w(0)=w(1)=0. $$ This inhomogeneous equation can be solved using the eigenfunctions of $L$ with zero endpoint conditions.