On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop

Abstract

Let y 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$H_\gamma = - \tfrac{{d^2 }}{{ds^2 }} + \kappa ^2 (s)$ acting on the space of square integrable 2π-periodic functions. A natural conjecture is that the lowest spectral value e₀ (y) of H_y is bounded below by 1 for any y (this value is assumed when y is a circle). We study a family of curves y that includes the circle and for which e₀(y) = 1 as well. We show that the curves in this family are local minimizers, i.e., e₀(y) can only increase under small perturbations leading away from the family. To our knowledge, the full conjecture remains open.