Lieb–Thirring and Jensen sums for non-self-adjoint Schrödinger operators on the half-line

We prove upper and lower bounds for sums of eigenvalues of Lieb–Thirring type for non-self-adjoint Schrödinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases.