Green function of discontinuous boundary-value problem with transmission conditions

In this paper, we deal with Sturm-Liouville-type problems when the potential of the differential equation may have discontinuity at one inner point and the eigenparameter appears not only in the differential equation, but also in both boundary and transmission conditions. By modifying some techniques of (Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edin. 1977; 77A:293-308; Eigenfunction Expenses Associated with Second-Order Differential Equations I (2nd edn). Oxford University Press: London, 1962) we generalize some results of the classic regular Sturm-Liouville problems. In particular, we construct Greenʹs function, and derive asymptotic approximation formulae for Greenʹs function. Further, we introduce a new operatortheoretic formulation in suitable Hilbert space such a way that the considered problem can be interpreted as the eigenvalue problem of this operator and constract the resolvent of this operator in terms of Greenʹs function. Finally, we estimate the norm of resolvent of this operator.