Degenerate boundary conditions on a geometric graph (Q2313253)

The paper studies degenerate boundary conditions on a simple graph with Sturm-Liouville operator, i.e. the cases when the spectrum is either empty or the whole \(\mathbb{C}\). It has been proven by \textit{M. H. Stone} [Trans. A. M. S. 29, 23--53 (1927; JFM 53.0429.01)] that for a symmetric potential \(q(x) = q(1-x)\) on an interval \((0,1)\) the spectrum of the problem \[ \begin{aligned} -y''(x) + q(x) y(x) & = \lambda y(x),\\ y(0)\pm y(1) & = 0,\\ y'(0)\mp y'(1) & = 0 \end{aligned} \] covers the whole complex plane. In the current paper, the authors consider a star graph with three edges and the problem given by \[ \begin{aligned} - y_i''(x) + q_i(x) y_i(x) & = \lambda y_i(x),\\ y_1(0) = y_2(0) & = y_3(0),\\ y_1'(0) + y_2'(0) + y_3'(0) & = 0,\\ a_{i1} y_1(\ell_1)+a_{i2} y_2(\ell_2)+a_{i3} y_3(\ell_3)+a_{i4} y_1'(\ell_1)+a_{i5} y_2'(\ell_2)+a_{i6} y_3'(\ell_3)& = 0 \end{aligned} \] with given real potentials \(q_i \in L_1(0,\ell_i)\), \(i=1,2,3\) and complex constants \(a_{ij}\). The authors prove that for the edges of different lengths the mentioned problem has no degenerate boundary conditions. If all the lengths are the same and \(q_i(x) = q(x)\), for all \(i\), the empty spectrum is not possible, while the spectrum fills the whole complex plane for an infinite number of combinations of constants \(a_{ij}\).