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We prove an upper bound on the sum of the distances between the eigenvaluesof 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 withsingle-well potentials. We rely on a variation of the well-known commutationmethod. In the P\"oschl-Teller and Coulomb cases we are able to use theexplicit factorisations to establish improved bounds.

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