Sturm Liouville problem with additional term. Imagine you want to solve an ODE on $[a,b] \subset \mathbb{R}$
$f''(x) + (A(x) + B(x))f(x) = \lambda_n f(x)$, where $A,B$ are some smooth functions and $\lambda_n$ the n-th eigenvalue.
Furthermore, we know one solution $f_1$ to this ODE with a corresponding eigenvalue $\lambda_1$.
Also we are able to solve $g''(x) + A(x) g(x) = \lambda'_n g(x)$ completely ( we get all eigenvalues and eigenfunctions). Does this mean, that we can also solve$f''(x) + (A(x) + B(x))f(x) = \lambda_n f(x)$ for all eigenvalues and eigenfunctions? Or can we at least construct some more solution to this latter problem from what we know so far?
 A: For each $\lambda \in\mathbb{C} $, there are two linearly independent solutions of $f''+(A+B)f=\lambda f$ on $[a,b]$, provided $A$ and $B$ are well-behaved on $[a,b]$. So it makes no sense to talk about the $n$-th eigenvalue $\lambda_{n}$ or the n-th eigenfunction.
By the way, most people prefer to write $-\frac{d^{2}}{dx^{2}}f+(A+B)f=\lambda_{n}f$ so that the eigenvalues may be arranged as $\lambda_{1} < \lambda_{2} < \lambda_{3} < \cdots$, at least for separated endpoint conditions. This is better correlated with Quantum Mechanics, too. For large $n$, the eigenvalues behave asymptotically the same as for the case where $A+B=0$ (you can see this because $\lambda$ overwhelms $A+B$ in $-f''+(A+B-\lambda)f=0$.) The corresponding eigenfunctions are also asymptotic to the classical trigonometric solutions of $-f''-\lambda f=0$. This is so strongly true that the Fourier series for the perturbed series converges at a point $x$ iff the classical series converges at the same point, and the rate of convergence is comparable as well. So $A$ and $B$ don't make much different for the large eigenvalues. It is only the lowest eigenvalues and eigenfunctions which are strongly affected by the presence of the perturbation $B$. A Rayleigh-Ritz type of method where you look at $(f''+(A+B)f,f)/(f,f)$ for functions $f$ constrained to satisfy the desired endpoint conditions reveals a definite dependency on $B$ in this case, and it also helps you estimate the change in eigenvalues. There is more stability for regular problems on finite intervals that one might expect, at least for separated endpoint conditions and smooth, bounded coefficients $A$, $B$.
