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

Abstract

Let C be a smooth closed curve of length 2 Pi in R^3, and let k(s) be its curvature, regarded as a function of arc length. We associate with this curve the one-dimensional Schrodinger operator H_C = -d^2/ds^2 + k^2 acting on the space of square integrable 2 Pi - periodic functions. A natural conjecture is that the lowest spectral value e(C) is bounded below by 1 for any C (this value is assumed when C is a circle). We study a family of curves {C} that includes the circle and for which e(C)=1 as well. We show that the curves in this family are local minimizers, i.e., e(C) can only increase under small perturbations leading away from the family. To our knowledge, the full conjecture remains open.