Keller-Lieb-Thirring inequalities for Schrodinger operators on cylinders

This note is devoted to Keller-Lieb-Thirring spectral estimates for Schrodinger 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: this is a symmetry result. 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: this provides a symmetry breaking result. The optimal threshold between the two regimes is established in the case of the product of a sphere by a line. (C) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.