Spectral properties of Sturm-Liouville operators with discontinuities at finite points

Purpose: In this paper, we investigate a class of Sturm-Liouville operators with eigenparameter- dependent boundary conditions and transmission conditions at finite interior points. Methods: By modifying the inner product in a suitable Krein space K associated with the problem, we generate a new self-adjoint operator A such that the eigenvalues of such a problem coincide with those of A. Results: We construct its fundamental solutions, get the asymptotic formulae for its eigenvalues and fundamental solutions, discuss some properties of its spectrum, and obtain its Green function and the resolvent operator. Conclusions: Three important conclusions can be drawn: (1) the new operator A is self-adjoint in the Krein space K; (2) if θi 0, i = 1,m, and ρj 0, j = 1, 2, then, the eigenvalues of the problem (Equations 1 to 5) are analytically simple; (3) the residual spectrum of the operator A is empty, i.e., σr(A) = ∅.