A Sturm separation theorem for a linear \(2n\)-th order self-adjoint differential equation (Q1890818)

This article primarily concerns \(n\)-fold zeros \(z\) of solutions \(y\) (i.e. \(y^{(j)}(z)= 0\) for every \(j=0,\dots,n-1\)) to even order linear DE of type (1) \(y^{(2n)}= py\) in an interval \(I= [x_0,\infty)\), where \(p\in C(I,\mathbb{R})\). If (1) has a nontrivial solution with \(n\)-fold zeros at points \(b\) and \(c\), \(x_0<b<c\), the main theorem states, for any \(a\in[x_0,b)\), there exists a solution \(v\) to (1) and \(d\in[b,c]\) such that \(v\) has \(n\)-fold zeros at both \(a\) and \(d\). Preliminary results include a matrix analogue of Green's formula and Wronskian formulas of the Weyl-Kodaira and \textit{W. N. Everitt} types [Q. J. Math., Oxf. II. Ser. 10, 145-155 (1959; Zbl 0087.29501)]. The proofs are based on Sturm-Picone type identity and analysis of \textit{K. Kreith} [Lect. Notes Math. 324, Springer (1973; Zbl 0258.35001)].