Inverse eigenvalue problems for nonlocal Sturm-Liouville operators on a star graph.(English)Zbl 1249.34044

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 $$\bigoplus_{j=1}^m L_2(0,l_j)$$, $(A(\psi_1,\ldots ,\psi_m))(x)=(-\psi_1''(x)+v_1(x)\psi_1(0),\ldots , -\psi_m''(x)+v_m(x)\psi_m(0)).$ Its domain consists of $$(\psi_1,\ldots ,\psi_m)\in \bigoplus_{j=1}^m W^2_2(0,l_j)$$ satisfying the boundary conditions $\psi_j(l_j)=0,\;j=1,\ldots ,m;\quad \psi_1(0)=\psi_2(0)=\ldots =\psi_m(0);$
$\sum\limits_{j=1}^m \left[ \psi_j'(0)- \int\limits_0^{l_j}\psi_j(x)\overline{v_j(x)}\,dx\right] =0,$ $$v_j$$ are given functions.
The 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 $$v_j$$.

MSC:

 34A55 Inverse problems involving ordinary differential equations 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47E05 General theory of ordinary differential operators 34L05 General spectral theory of ordinary differential operators 34B45 Boundary value problems on graphs and networks for ordinary differential equations