Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders
Jean Dolbeault, Maria J. Esteban, Michael Loss · Comptes Rendus Mathématique · 2015
This note is devoted to Keller–Lieb–Thirring spectral estimates for Schrödinger 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.