Keller-Lieb-Thirring inequalities for Schr\"odinger operators on cylinders

This note is devoted to Keller-Lieb-Thirring spectral estimates for Schr"odinger operators on infinite cylinders: the absolute value of the ground state level is bounded by a function of a norm of the potential. Optimal potentials with small norms are shown to depend on a single variable. The proof is a perturbation argument based on recent rigidity results for nonlinear elliptic equations on cylinders. Conversely, optimal single variable potentials with large norms must be unstable. The optimal threshold between the two regimes is established in the case of the product of a sphere by a line.