It is well known that in order to determine the coefficients of a Sturm-Liouville problem on a finite interval with Dirichlet or Robin boundary conditions, the spectrum alone is not enough and thus one needs some extra information like a sequence of norming constants or a second spectrum. K. Mochizuki and I. Trooshin [J. Inverse Ill-Posed Probl. 9, No. 4, 425–433 (2001; Zbl 1035.34008)] proved that the values of the logarithmic derivative of eigenfunctions at the midpoint of the interval can also be used as this extra information. This result was later generalized to the case of boundary conditions depending polynomially on the eigenvalue parameter [Y. P. Wang, Result. Math. 63, No. 3–4, 1131–1144 (2013; Zbl 1279.34027); N. J. Guliyev, Ann. Mat. Pura Appl. (4) 199, No. 4, 1621–1648 (2020; Zbl 1448.34064)].
The paper under review is devoted to similar questions for problems with boundary conditions depending linearly on the square root of the eigenparameter.
Reviewer’s remarks: The paper is very poorly written. As their main tool (Lemma 2.1) they cite a result which is devoted to the case of boundary conditions depending polynomially on the eigenvalue parameter (not its square root) and without any weight function. Some values of the constants in the boundary conditions should explicitly be excluded, since otherwise it is even possible that there are no eigenvalues at all. The readers need themselves to guess how the eigenvalues are indexed, and so on.
Most importantly, nothing is said about whether one actually needs that much spectral data. Although the spectral theory of problems with boundary conditions depending polynomially on the eigenvalue parameter is very similar to the classic case (i.e., Dirichlet or Robin boundary conditions), the situation changes drastically as soon as the square root of the eigenparameter enters one of the boundary conditions. In the latter case, the spectrum alone determines the boundary value problem. This can be explained by the fact that the knowledge of this spectrum is actually equivalent to the knowledge of the spectra of two problems of the former type (see [H. Hochstadt, Acta Math. 119, 173–192 (1967; Zbl 0155.13002)] for details). Therefore, in the reviewer’s opinion, the authors first need to clarify whether the spectrum alone determines their boundary value problem and to proceed to the problems considered in the paper only if the answer to this question is negative.