, Volume 15, Issue 4, pp 543-563

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

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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 Hy 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.

Communicated by Elliot Lieb