We consider a Sturm-Liouville equation ℓy := −y ′′ + qy = λy on the intervals (−a, 0) and (0, b) with a, b > 0 and q ∈ L 2 (−a, b). We impose boundary conditions y(−a) cos α = y ′ (−a) sin α, y(b) cos β = y ′ (b) sin β, where α ∈ [0, π) and β ∈ (0, π], together with transmission conditions rationally-dependent on the eigenparameter viawith bi, aj > 0 for i = 1, . . . , N, and j = 1, . . . , M . Here we take η, κ ≥ 0 and N, M ∈ N0. 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) C N * C M * , for suitable N * , M * , is given. The Green's function and the resolvent of the related Hilbert space operator are expressed explicitly.Mathematics Subject Classification (2010). Primary 34B24; Secondary 34A36, 34B07.

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