A characterization of some even vector-valued Sturm–Liouville problems

We call “even” a Sturm–Liouville problem
\begin{gather} -y''+Q(x)y=\lambda y, \quad 0\leq x\leq\pi, \tag{1} \\ y'(0)-hy(0)=0, \tag{2} \\ y'(\pi)+Hy(\pi)=0, \tag{3} \end{gather}
in which $H=h$ and $Q(\pi-x)\equiv Q(x)$ on $[0,\pi]$. In this paper we study the vector-valued case, where the potential $Q(x)$ is a real symmetric $d\times d$ matrix for each $x$ in $[0,\pi],$ and the entries of $Q$ and their first derivatives (in the distribution sense) are all in $L^2[0,\pi]$. We assume that $h$ and $H$ are real symmetric $d\times d$ matrices.
We prove that a vector-valued Sturm–Liouville problem (1)–(3) is even if and only if, for each eigenvalue $\lambda$, whose multiplicity is $r=r_{\lambda}$ (where $1\le r\le d$, and where $\varphi_1(x,\lambda),\dots,\varphi_r(x,\lambda)$ denote orthonormal eigenfunctions belonging to $\lambda$), there exists an $r\times r$ matrix $A=(a_{ij})$ (which may depend on $\lambda$ and on the choice of basis $\{\varphi_i(x,\lambda)\}_{i=1}^r$, but does not depend on $x$) such that
(1) A is orthogonal and symmetric, and
(2) for $1\le i\le r$, $\varphi_i(\pi,\lambda)=\sum_{j=1}^ra_{ij}\varphi_j(0,\lambda)$.
\noindent To some extent our theorem can be considered a generalization of N. Levinson's results in [2].