Asymptotics for eigenvalues of Schrödinger operator with small translation and Dirichlet condition

We consider a non-self-adjoint Schrödinger operator on the unit interval with Dirichlet conditions perturbed by an operator of small translation. The main result is a three-term asymptotic expansion for the eigenvalues with respect to their index, and this asymptotics is uniform in the small translation. We also show that the system of eigenfunctions and generalized eigenfunctions of the considered operators forms a Bari basis in the space of square integrable functions on the considered unit interval.