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

Abstract

We consider a Sturm-Liouville equation $\ell y:=-y^{\prime \prime }+qy=\lambda y$ on the intervals $(-a,0)$ and $(0,b)$ with $a,b>0$ and $q\in L^2(-a,b)$. We impose boundary conditions $y(-a)\cos \alpha =y^{\prime }(-a)\sin \alpha$, $y(b)\cos \beta =y^{\prime }(b)\sin \beta$, 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(λη-ξ-\sum\limits_{i=1}^{N} \frac{b_i^2}{λ-c_i}\right) &= y'(0^+) - y'(0^-),\\ y'(0^-)\left(λκ+ζ-\sum\limits_{j=1}^{M}\frac{a_j^2}{λ-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 $N,M\in {\mathbb{N}}_0$. 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 $L^2(-a,b)\bigoplus \C^{N^*} \bigoplus \C^{M^*}$, for suitable $N^{\ast },M^{\ast }$, is given. The Green's function and the resolvent of the related Hilbert space operator are expressed explicitly.