On Recovering Sturm--Liouville-Type Operator with Delay and Jump Conditions

In this manuscript, the second order differential operators with constant delay and transmission boundary conditions are studied. The asymptotic forms of the characteristic functions and eigenvalues  of the operators are obtained. Therefore, an inverse spectral problem of recovering the operators using the spectra  of two different    Dirichlet-Dirichlet and  Dirichlet-Neumann  problems, is investigated. For this purpose, to prove the uniqueness of the potential, the integral equations of the potential function are formulated and then utilized. Indirectly, by calculating the coefficients of  Fourier series,  the potential function is reconstructed.