Sturm-Liouville problems with transfer condition Herglotz dependent on the eigenparameter -- Hilbert space formulation

Abstract

We consider a Sturm-Liouville equation $\ell y:=-y'' + qy = \lambda y$ on the intervals $(-a,0)$ and $(0,b)$ with $a,b>0$ and $q \in L^2(-a,b)$. We impose boundary conditions $y(-a)\cos\alpha = y'(-a)\sin\alpha$, $y(b)\cos\beta = y'(b)\sin\beta$, where $\alpha \in [0,\pi)$ and $\beta \in (0,\pi]$, together with transmission conditions rationally-dependent on the eigenparameter via \begin{align*} -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{align*} with $b_i, a_j>0$ for $i=1,\dots,N,$ and $j=1,\dots,M$. Here we take $\eta, \kappa \ge 0$ and $N,M\in \N_0$. 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 $L^2(-a,b)\bigoplus \C^{N^*} \bigoplus \C^{M^*}$, for suitable $N^*,M^*$, is given. The Green's function and the resolvent of the related Hilbert space operator are expressed explicitly.