On an Isoperimetric Inequality for a Schrödinger Operator Depending on the Curvature of a Loop

Abstract Let γ be a smooth closed curve of length 2π in R3, and let κ(s) be its curvature, regarded as a function of arc length. We associate with this curve the one-dimensional Schrödinger operator Hγ = − d 2 ds + κ 2(s) acting on the space of square integrable 2π-periodic functions. A natural conjecture is that the lowest spectral value e0(γ) of Hγ is bounded below by 1 for any γ (this value is assumed when γ is a circle). We study a family of curves {γ} that includes the circle and for which e0(γ) = 1 as well. We show that the curves in this family are local minimizers; i.e., e0(γ) can only increase under small perturbations leading away from the family. To our knowledge, the full conjecture remains open.