On factorizations of selfadjoint ordinary differential operators

Abstract:

Consider an ordinary linear differential operator $L$, of order $n \geqslant 1$, represented by $Lu \equiv {a_n}(t){u^{(n)}} + \cdots + {a_0}(t)u\;\forall u \in {C^n}(a,b)$, with real-valued coefficients ${a_k} \in {C^k}(a,b)$, $0 \leqslant k \leqslant n$, ${a_n} \ne 0$ on $(a,b)$. According to a classical result, if $L$ is formally selfadjoint on $(a,b)$ then it has a factorization of the type $Lu \equiv {p_n}[{p_{n - 1}}( \cdots ({p_1}({p_0}u)’)’ \cdots )’]’\forall u \in {C^n}(a,b)$, where the ${p_k}$’s are sufficiently-smooth and everywhere nonzero functions on $(a,b)$ such that ${p_k} = {p_{n - k}}$ $(k = 0, \ldots ,n)$. In this note we shall examine this result critically and show by means of counterexamples that the different classical proofs are either merely local or purely heuristic. A proof, which is both rigorous and global, is inferred from recent results on canonical factorizations of disconjugate operators. In addition, information is obtained on the behavior of the ${p_k}$’s at the endpoints of $(a,b)$ which may prove useful in applications.