Discontinuous Sturm-Liouville Operator With Boundary Condition Depending On Spectral Parameter And Indefinite Weight Function

The paper studies the discontinuous Sturm-Liouville operator L with boundary con-dition containing spectral parameter and indefinite weight function. Firstly, a Krein space K and a new operator A related to the boundary-value problem are constructed to make the eigenvalues of the operators A and L same. It is proved that the operator A is self-adjoint in the K. Secondly, we also construct a Hilbert space H related to K and self-adjoint operator S in it. By using spectrum theory of self-adjoint operator in Krein space and the spectral properties of operator S, we come to the conclusion that A has only real point spectrum and they are unbounded from below and from above. Based c the the spectrum curve theory, we prove that the eignvalues of A have no finite cluster point and can be indexed. Thus the eignvalues of the boundary value problem are real, and they are unbounded from below and from above, have no finite cluster point, and can be indexed to satisfy the inequalities as following:Thirdly, by studying the operator L itself, we obtain thatλis the necessary and suffi-cient condition for its eigenvalue, and then we construct the Green's function of the new operator A. Finally, a specific example is given to get the eigenvalues distribution.