Inverse problem for the differential pencil on an arbitrary graph with partial information given on the coefficients (Q2009456)

The inverse spectral problem is studied for the differential equations \[ -y_j''(x_j)+(q_j(x_j)+2\lambda p_j(x_j)-\lambda^2)y_j(x_j)=0,\; j=\overline{1,m}, \] on an arbitrary compact geometrical graph with \(m\) edges. The inverse problem consists in recovering the potentials on one edge from a part of the spectrum provided that the potentials are known on the remaining part of the graph. The main results of the paper are the uniqueness theorem and a constructive procedure for the solution of this inverse problem.