I have the spectral problem $X'' + (\alpha(x) - \lambda)X = 0$, with boundary conditions $X(0) = 0 = X(1)$. This is a Sturm Liouville problem, and the book I am reading says that clearly
$$ ||\alpha(x)||_{L^{\infty}(0,1)} \geq \lambda_1 > \lambda_2 \ldots $$
which I think makes sense, since we want non-trivial solutions, right?
But what about the ODE
$$ X'' + \alpha(x)X' - \lambda X = 0 $$
with the same boundary conditions? How would we go about knowing the relation between $\alpha(x)$ and the eigenvalues?