Regularity of eigenfunctions of an ordinary differential operator

Question

Let $I \subseteq \mathbb{R}$ be an open interval and $T:D(T)\subseteq L^2(I) \to L^2(I)$ a differential operator given by $$(Tf)(x):= \sum_{j=0}^n a_j(x)f^{(j)}(x), \quad f\in D(T), \ x \in I,$$ where the derivatives are understood in the weak sense. Under what conditions in $a_j(x)$ can we say that $\lambda \in \mathbb{C}$ is an eigenvalue of $T$ iff there exists a non-trivial classic solution $f \in D(T)$ of $(T-\lambda)f=0$?. Do you know any reference for that?