Boundary values of generalized solutions of a homogeneous Sturm–Liouville equation in a space of vector functions

We consider a differential equation of the form $-y''+A^2y=0$, where $A$ is a self-adjoint operator in a Hilbert space $H$. We show that each generalized solution of this equation inw $W_{-m}(0,b)$ ($0<b<\infty$, $m\ge0$) has boundary values in the space $H_{-m-1/2}$, where $H_j$ ($-\infty<j<\infty$) is the Hilbert scale of spaces generated by the operator $A$, and $W_{-m}(0,b)$ is the space of continuous linear functionals on order $\mathring W_m(0,b)$, the completion of the space of infinitely differentiable vector functions with compact support with respect to the norm $\|u\|_{W_m(0,b)}=(\|u\|_{L_2(H_m,(0,b))}+\|u\|_{L_2(H,(0,b))}^{(m)})$. It follows that each function $u(t,x)$ which is harmonic in the strip $G=[0,b]\times(-\infty,\infty)$ and which is in the space that is dual to order $\mathring W_2^m(G)$ has limiting values as $t\to0$ and $t\to b$ in the space $W_2^{-m-1/2}(-\infty,\infty)$.