Compatibility conditions for two first-eigenvalue functions

Characterize the complete set of compatibility conditions that two first eigenvalue functions $\lambda(t,r_1)$ and $\lambda(t,r_2)$, associated with distinct coupling constants $r_1\ne r_2$ in the multi-point perturbed Dirac-weighted Sturm–Liouville problem, must satisfy in order to arise from the same potential $q$.

Background

To address the underdetermination in the algebraic reconstruction system for $n\ge 3$, the authors propose using two first eigenvalue functions corresponding to distinct coupling constants $r_1$ and $r_2$. The resulting augmented system contains more equations than unknowns.

The additional equations may impose nontrivial restrictions on the pair of spectral data functions. The paper leaves unresolved the problem of identifying exactly which compatibility conditions are necessary and sufficient for both functions to originate from one common potential $q$.