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

Анотація

We call "even" a Sturm-Liouville problem$$-y'' +Q(x)y= \lambda y, 0\le x\le \pi,\qquad \;\;(1) $$ $$y'(0) - hy(0)= 0, \qquad \qquad \qquad \quad \quad (2) $$ $$y'(\pi) + Hy(\pi)= 0 \qquad \qquad \qquad \quad \quad (3) $$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 $Q(x)$ 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),\ldots ,\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 $1 \le i \le r$, $\varphi_i(\pi,\lambda)=\sum_{j=1}^ra_{ij}\varphi_j(0,\lambda).$ To some extent our theorem can be considered a generalization of N. Levinson's results in [2].