Situation (cut short): Corresponding solutions (by eigenvalue) of two given regular Sturm-Liouville problems with homogeneous Neumann BC, same spectrum but possibly distinct coefficient functions, "happen to" match on an open subset of their domain (an interval).
- Do corresponding solutions match on the whole domain? (identity principle/unique continuation)
- If not, is there an additional criterion or regularity assumption that would allow this conclusion?
Detailed formulation:
Assume the SL-problems are defined on an interval $(a_0, a_1)$ and are written for $j \in ${$1, 2$} as \begin{equation} \tag{1} \frac {\mathrm {d} }{\mathrm {d} x}\left[p_j(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q_j(x)y_{j, i}=-\lambda_{j, i} w_j(x)y_{j, i} \end{equation} with $\lambda_{1, i} = \lambda_{2, i}$ for all $i$. Boundary condition: \begin{equation} \tag{2} \frac {\mathrm {d} }{\mathrm {d} x} y_{j, i} = 0 \end{equation} We further have for all $i$ \begin{equation} \tag{3} y_{1, i}(x)=y_{2, i}(x) \qquad \forall x \in (b_0, b_1) \subset (a_0, a_1) \end{equation}
Question: Can we conclude \begin{equation} \tag{4} y_{1, i}(x)=y_{2, i}(x) \qquad \forall x \in (a_0, a_1) \end{equation}?
If it helps, I am primarily interested in the case $q_j = 0$. $w_j, p_j > 0$ can be assumed everywhere. Assume that the coefficient functions are regular enough to let $y_{1, i}$ and $y_{2, i}$ have the unique continuation property each (e.g. analytic coefficient functions but not necessarily).
It appears to me that this question may be related to the inverse SL problem. By that theory, solutions are determined for a prescribed spectrum, if Neumann and Dirichlet boundary conditions are given simultaneously (please correct me if I am wrong here). The described situation is somewhat like replacing the Dirichlet boundary condition by prescribed values on an open subinterval of the domain. However I think the inverse problem just asks for determining the solution, not for an identity principle w.r.t. a solution of a related other SL-problem.
Pointers to literature are particularly welcome. I found literature hard to search for the described question; please excuse if I overlooked something obvious or a duplicate of this question.
(Edit: ) Given the comments below I would like to refine the question as follows:
Can we choose the independent variable outside $(b_0,b_1)$ such that $y_{1,i}$ matches $y_{2,i}$?
Note: For the $q_j = 0$ case, let's say $i=0$ denotes the constant solution, i.e. $\lambda_{j, 0}=0$, and $i=1$ the first non-constant solution. Then $y_{j, 1}$ are monotonic functions. So for $i=1$ the question would boil down to whether $y_{1, 1}$ and $y_{2, 1}$ match on the boundary.
