Sturm-Liouville Operators With Transmission Conditions And Eigenparameter-Dependent Boundary Conditions

In this paper, we investigate the Sturm-Liouville operators with transmission conditions and eigenparameter-dependent boundary conditions. Firstly, By means of studying the operator itself, we investigate properties of its eigenvalues. We obtain the necessary and sufficient conditions for A is a eigenvalue, and prove that the eigenvalue of the boundary-value problem are bounded below, and they are countably infinit and can cluster only at∞. Secondly, by defmeding a self-adjoint linear operator A in a suitable Hilbert space H , we prove that the completeness of their eigenfunctions, finally, we construct the Green's function of the new operator A.