DG-MP-PDE Seminar: On an isoperimetric inequality for a Schroedinger operator depending on the curvature of a loop

Let \gamma be a smooth closed curve of length 2\pi in R^3, and let \kappa(s) be its curvature regarded as a function of arc length s. We associate with this curve the one-dimensional Schroedinger operator H_\gamma = -d^2/ds^2 + \kappa^2(s) acting on the space of square integrable 2\pi-periodic functions. A natural conjecture is that the lowest eigenvalue e_0(\gamma) of H_\gamma is bounded below by 1 for any \gamma (this value is assumed when \gamma is a circle). We study a family of curves which includes the circle and for which e_0(\gamma)=1 as well, and show that the curves in this family are local minimizers; i.e., e_0(\gamma) does not decrease under small perturbations. A connection between the inequality and a dynamical elastica will be described. The conjecture remains open.