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

TL;DR

This paper studies a Sturm-Liouville problem with transfer conditions depending rationally on the eigenparameter, providing a Hilbert space formulation, eigenvalue multiplicity analysis, and explicit Green's function and resolvent expressions.

Contribution

It introduces a novel eigenvalue problem with transfer conditions rationally dependent on the eigenparameter and develops a Hilbert space framework for its analysis.

Findings

Eigenvalues' geometric multiplicity is characterized.

Conditions for eigenvalues to have multiplicity two are identified.

Explicit formulas for Green's function and resolvent are derived.

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.