Inverse problem for differential pencils on a hedgehog graph (Q309821)

A hedgehog graph \(G\) is composed of a cycle with \(N\) vertices (and therefore \(N\) edges) and \(r_j\) (\(j=1,\dots,N\)) spines at each of the \(N\) vertices. With \(r=r_1+\dots+r_N\) it follows that the graph \(G\) has \(r+N\) edges \(e_j\) and \(r+N\) vertices \(v_j\), where exterior edges and vertices carry the indices \(j=1,\dots,r\). A function \(Y\) on \(G\) is represented as \(Y=(y_j)_{j=1}^{r+N}\) with \(y_j\) being a function on \(e_j\) with independent variable \(x_j\in[0,T_j]\) and \(x_j=0\) representing the exterior vertex for \(j=1,\dots,r\). With the eigenvalue parameter \(\rho \) consider on \(G\) the differential equation \[ y_j^{\prime\prime}(x_j)+(\rho ^2+\rho p_j(x_j)+q_j(x_j))y_j(x_j)=0,\quad x_j\in[0,T_j],\quad j=1,\dots, r+N,\tag{1} \] where \(p_j\) and \(q_j\) are complex-valued functions on \([0,T_j]\). Let \[ U_j(Y)=y_j'(0)-(i\rho h_j'+h_j)y_j(0),\quad j=1,\dots,r+N ,\tag{1a} \] where \(h_j, h_j'\in \mathbb{C}\) and \(h_j'\neq \pm1\) for \(j=1,\dots,r\). The boundary value problem \(B_0\) consists of (1) together with the boundary conditions (1a) at the external vertices, \(j=1,\dots,r\), and with matching conditions at the internal vertices \(v_j\), \(j=r+1,\dots,r+N\). For \(p=1,\dots,r\) and \(1\leq \nu_1<\dots<\nu_p\leq1\), the boundary value problem \(B_{\nu_1,\dots,\nu_p}\) is derived from \(B_0\) by replacing \(U_k(Y)=0\) with \(y_k(0)=0\) for \(k=\nu_1,\dots,\nu_p\). For each of the interior vertices choose an exterior edge emanating from it. The index set of these exterior edges is denoted by \(\xi \). Also a sequence \(\Omega \) is needed, for which the author refers to two of his earlier publications. The main result of the paper is the solution the following inverse problem. Let \(\Omega \), \(\xi \) and the eigenvalues, counted with multiplicity, of the problems \(B_j\), \(j=0,\dots,r\), and \(B_{\nu_1,\dots,\nu_p}\), \(p=1,\dots,N\), \(1\leq \nu_1<\dots<\nu_p\leq r\), \(\nu_j\in \xi \) be given. Then \(p_j\), \(q_j\), \(h_j\) and \(h_j'\) are uniquely determined, and an algorithm for the recovery of those parameters is provided.