Sturm-Liouville Problems with Transfer Condition Herglotz Dependent on the Eigenparameter: Hilbert Space Formulation

We consider a Sturm-Liouville equation ly := -y '' + qy = lambda y on the intervals (-a, 0) and (0, b) with a, b > 0 and q is an element of L-2 (-a, b). We impose boundary conditions y(-a) cos alpha = y' (-a) sin alpha, y(b) cos beta = y' (b) sin beta, where alpha is an element of [0, pi) and beta is an element of (0, pi], together with transmission conditions rationally-dependent on the eigenparameter via - y(0(+)) (lambda eta - xi - Sigma(N)(i=1) b(i)(2)/lambda - c(i)) = y' (0(+)) - y' (0(-)), y'(0(-)) (lambda kappa - zeta - Sigma(M)(j=1) a(j)(2)/lambda - d(j)) = y (0(+)) - y (0(-)), with b(i), a(j) > 0 for i = 1,..., N, and j = 1,..., M. Here we take eta, kappa >= 0 and N, M is an element of N-0. The geometric multiplicity of the eigenvalues is considered and the cases in which the multiplicity can be 2 are characterized. An example is given to illustrate the cases. A Hilbert space formulation of the above eigenvalue problem as a self-adjoint operator eigenvalue problem in L-2 (-a, b) circle plus C-N* circle plus C-M*, for suitable N*, M*, is given. The Green's function and the resolvent of the related Hilbert space operator are expressed explicitly.