Spectral asymptotics for Schrodinger operator perturbed by translation operator

We consider the one-dimensional Schrödinger operator on the unit segment with Dirichlet condition and perturb it by a translation operator. The main result describes the asymptotics of the eigenvalues of this operator with respect to the index counting the eigenvalues, the resulting asymptotics is uniform in the translation. In the asymptotics, we explicitly find the terms generated by the translation operator. We establish that the system of eigenfunctions and generalized eigenfunctions of the considered operator forms a Bari basis in the space of functions square integrable on the unit segment.