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We consider a Sturm–Liouville equation (ell y:=-yu0027u0027 + qy = lambda y) on the intervals ((-a,0)) and (0, b) with (a,bu003e0) and (q  L^2(-a,b)). We impose boundary conditions (y(-a)cos alpha = yu0027(-a)sin alpha ), (y(b)cos beta = yu0027(b)sin beta ), where (alpha  [0,pi )) and (beta  (0,pi ]), together with transmission conditions rationally-dependent on the eigenparameter via $$begin{aligned} -y(0^+)left( lambda eta -xi -sum limits _{i=1}^{N} frac{b_i^2}{lambda -c_i}right)u0026= yu0027(0^+) - yu0027(0^-), yu0027(0^-)left( lambda kappa +zeta -sum limits _{j=1}^{M}frac{a_j^2}{lambda -d_j}right)u0026= y(0^+) - y(0^-), end{aligned}$$with (b_i, a_ju003e0) for (i=1,dots ,N,) and (j=1,dots ,M). Here we take (eta , kappa ge 0) and (N,Min {mathbb 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)bigoplus {mathbb C}^{N^*} bigoplus {mathbb C}^{M^*}), for suitable (N^*,M^*), is given. The Green’s function and the resolvent of the related Hilbert space operator are expressed explicitly.

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