A Faber–Krahn inequality for solutions of Schrödinger’s equation

Abstract

We consider nontrivial solutions of −Δu(x)=V(x)u(x), where u≡0 on the boundary of a bounded open region D⊂Rn, and V(x)∈L∞(D). We prove a sharp relationship between ‖V‖∞ 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 Schrödinger equation −Δu(x)=V(x)u(x).