  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.
