on the operator sturm-liouville problem with unbounded operator coefficients in boundary condition

In this study, we carry out the spectral analysis of operators generated by the sturm-liouville equation and boundary conditions, where both contain unbounded operator coefficients and a spectral parameter. such problems model many processes in classical and quantum mechanics as well as in mechanical engineering. in this paper, we provide a complete description of the domains corresponding to the problem operators in the exit space, a characterization of the spectrum, and the asymptotic behavior of the discrete spectrum. the theoretical results obtained in this study are particularly applicable to areas such as vibration analysis, stability, and mathematical modeling of dynamic systems in mechanical engineering. the study of the spectral characteristics of operators provides an important theoretical basis for determining the natural frequencies and critical loads of thin-walled structures with complex boundary conditions. furthermore, it provides a general mathematical framework that can be used to model the behavior of engineering systems under thermoelastic, piezoelectric, and magnetoelastic effects. in these respects, the study has the potential to contribute to structural safety, vibration control, and advanced material-based design approaches in mechanical engineering.