Inverse eigenvalue problems for nonlocal Sturm-Liouville operators on a star graph (Q2896662)

The nonlocal operator on a star graph with center vertex at the origin and \(m\) edges of lengths \(l_j\), \(j=1,2,\ldots ,m\), is defined as follows. The operator \(A\) acts on the space \(\bigoplus_{j=1}^m L_2(0,l_j)\), NEWLINE\[NEWLINE (A(\psi_1,\ldots ,\psi_m))(x)=(-\psi_1''(x)+v_1(x)\psi_1(0),\ldots , -\psi_m''(x)+v_m(x)\psi_m(0)). NEWLINE\]NEWLINE Its domain consists of \((\psi_1,\ldots ,\psi_m)\in \bigoplus_{j=1}^m W^2_2(0,l_j)\) satisfying the boundary conditions NEWLINE\[NEWLINE \psi_j(l_j)=0,\;j=1,\ldots ,m;\quad \psi_1(0)=\psi_2(0)=\ldots =\psi_m(0); NEWLINE\]NEWLINE NEWLINE\[NEWLINE \sum\limits_{j=1}^m \left[ \psi_j'(0)- \int\limits_0^{l_j}\psi_j(x)\overline{v_j(x)}\,dx\right] =0, NEWLINE\]NEWLINE \(v_j\) are given functions.NEWLINENEWLINEThe author gives a complete description of the spectrum of the operator \(A\) and presents an algorithm of solving the inverse problem of finding, knowing the eigenvalues and their multiplicities, all the problem data -- the number \(m\), the lengths \(l_j\), and the local potentials \(v_j\).