reconstruction of discontinuous sturm-liouville pencils with the eigenvalue in the boundary condition on the half-line

This work considers the discontinuous differential pencil on the half-line with the spectral boundary condition. we establish some uniqueness theorems on the potentials for the sturm-liouville pencil by the incomplete inverse problem and the interior inverse problem methods. we determine the potentials by only a set of eigenvalues knowing the coefficients _0; _1; _2; _3 and potentials q0(x); q1(x) on (0; a): we also establish the potentials by a set of values of eigenfunctions at some internal point x = a and eigenvalues.