  Consider a one dimensional quantum mechanical particle described by the
Schroedinger equation on a closed curve of length $2\pi$. Assume that the
potential is given by the square of the curve's curvature. We show that in this
case the energy of the particle can not be lower than 0.6085. We also prove
that it is not lower than 1 (the conjectured optimal lower bound) for a certain
class of closed curves that have an additional geometrical property
