The Sturm–Liouville problem with a discontinuity condition

This paper deals with spectral analysis for the Sturm-Liouville problem with distribution potential of class $W_2^{-1}(0,T)$ and with a discontinuity inside the interval. The class of generalized functions $W_2^{-1}(0,T)$ includes, in particular, potentials with the Dirac $\delta$-function-type singularities and with the Coulomb singularities $1/{(x-c)}$, which are used in models of quantum mechanics. The Sturm-Liouville boundary value problems arise when solving equations of mathematical physics by variable separation method. They are widely used in various fields of mathematics, mechanics, physics, etc. The properties of the eigenvalues and eigenfunctions of the Sturm-Liouville problem play an important role in this case. Boundary value problems with discontinuities inside an interval are related to discontinuous properties of medium. In particular, such problems arise in geophysical models of the Earth and in radio electronics. A large number of studies are concerned with solving inverse spectral problems with discontinuity conditions. At the same time, spectral theory for the Sturm-Liouville operators with distribution potentials has been rapidly developing. Considering distribution coefficients in the inverse problem theory has led to spectral data characterization for new classes of differential operators. It is promising to transfer that approach to problems with discontinuities, which explains the interest to studying the discontinuous Sturm-Liouville problem with singular potential. The problem, which is considered in this paper, is essentially new. To the best of the authors' knowledge, this problem was not investigated before. This paper aims to study some spectral properties of the generalized Sturm-Liouville problem. In particular, we obtain the asymptotic formulas for its eigenvalues. Our methods are based on spectral theory of differential operators and on the theory of analytic functions. In the future, we plan to apply the obtained results to formulation and investigation of inverse spectral problems that consist in the recovery of the boundary value problem coefficients for spectral characteristics.