We consider a Sturm–Liouville equation $$\ell y:=-y'' + qy = \lambda y$$ ℓy:=-y′′+qy=λy on the intervals $$(-a,0)$$ (-a,0) and (0, b) with $$a,b>0$$ a,b>0 and $$q \in L^2(-a,b)$$ q∈L2(-a,b) . We impose boundary conditions $$y(-a)\cos \alpha = y'(-a)\sin \alpha $$ y(-a)cosα=y′(-a)sinα , $$y(b)\cos \beta = y'(b)\sin \beta $$ y(b)cosβ=y′(b)sinβ , where $$\alpha \in [0,\pi )$$ α∈[0,π) and $$\beta \in (0,\pi ]$$ β∈(0,π] , together with transmission conditions rationally-dependent on the eigenparameter via $$\begin{aligned} -y(0^+)\left( \lambda \eta -\xi -\sum \limits _{i=1}^{N} \frac{b_i^2}{\lambda -c_i}\right)&= y'(0^+) - y'(0^-),\\ y'(0^-)\left( \lambda \kappa +\zeta -\sum \limits _{j=1}^{M}\frac{a_j^2}{\lambda -d_j}\right)&= y(0^+) - y(0^-), \end{aligned}$$ -y(0+)λη-ξ-∑i=1Nbi2λ-ci=y′(0+)-y′(0-),y′(0-)λκ+ζ-∑j=1Maj2λ-dj=y(0+)-y(0-), with $$b_i, a_j>0$$ bi,aj>0 for $$i=1,\dots ,N,$$ i=1,⋯,N, and $$j=1,\dots ,M$$ j=1,⋯,M . Here we take $$\eta , \kappa \ge 0$$ η,κ≥0 and $$N,M\in {\mathbb N}_0$$ 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 $$L^2(-a,b)\bigoplus {\mathbb C}^{N^*} \bigoplus {\mathbb C}^{M^*}$$ L2(-a,b)⨁CN∗⨁CM∗ , for suitable $$N^*,M^*$$ N∗,M∗ , is given. The Green’s function and the resolvent of the related Hilbert space operator are expressed explicitly.