Lieb-Thirring type bounds for perturbed Schr\"odinger operators with single-well potentials

We prove an upper bound on the sum of the distances between the eigenvalues of a perturbed Schr\"odinger operator $H_0-V$ and the lowest eigenvalue of $H_0$. Our results hold for operators $H_0=-\Delta-V_0$ in one dimension with single-well potentials. We rely on a variation of the well-known commutation method. In the P\"oschl-Teller and Coulomb cases we are able to use the explicit factorisations to establish improved bounds.