A Faber-Krahn inequality for solutions of Schr?dinger's equation

摘要

We consider nontrivial solutions of -δu(x)=V(x)u(x), where u≡0 on the boundary of a bounded open region D?R ~n, and V(x)∈L ~∞(D). We prove a sharp relationship between {norm of matrix}V{norm of matrix} _∞ and the measure of D, which generalizes the well-known Faber-Krahn theorem. We also prove some geometric properties of the zero sets of the solution of the Schr?dinger equation -δu(x)=V(x)u(x).