Regularity of eigenfunctions of an ordinary differential operator

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?

• @Thomas thank you for your observation. I have edited my question. – Kanido mat Nov 15 at 18:39