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

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© 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©=1 as well. We show that the curves in this family are local minimizers, i.e., e© can only increase under small perturbations leading away from the family. To our knowledge, the full conjecture remains open.