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

$$\begin{aligned} -y(0^+)\left( \lambda \eta -\xi -\sum \limits _{i=1}^{N} \frac{b_i^2}{\lambda -c_i}\right)&= y'(0^+) - y'(0^-),\\ y'(0^-)\left( \lambda \kappa +\zeta -\sum \limits _{j=1}^{M}\frac{a_j^2}{\lambda -d_j}\right)&= y(0^+) - y(0^-), \end{aligned}$$

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 $L^2(-a,b)\bigoplus {\mathbb C}^{N^*} \bigoplus {\mathbb C}^{M^*}$, for suitable $N^*,M^*$, is given. The Green’s function and the resolvent of the related Hilbert space operator are expressed explicitly.