Sturm–Liouville problems with discontinuities at two points

In this paper we extend some spectral properties of regular Sturm–Liouville problems to those which consist of a Sturm–Liouville equation with piecewise continuous potentials together with eigenparameter-dependent boundary conditions and four supplementary transmission conditions. By modifying some techniques of [C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293–308; E. Tunç, O.Sh. Muhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput. 157 (2004) 347–355; O.Sh. Mukhtarov, E. Tunç, Eigenvalue problems for Sturm–Liouville equations with transmission conditions, Israel J. Math. 144 (2004) 367–380] and [O.Sh. Mukhtarov, M. Kadakal, F.Ş. Muhtarov, Eigenvalues and normalized eigenfunctions of discontinuous Sturm–Liouville problem with transmission conditions, Rep. Math. Phys. 54 (2004) 41–56], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulae for the eigenvalues and eigenfunctions.