A lower bound for the ground state energy of a Schrödinger operator on a loop

Abstract

Consider a one-dimensional quantum mechanical particle described by the Schrödinger equation on a closed curve of length 2 π 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 cannot be lower than 0.6085 0.6085 . We also prove that it is not lower than 1 1 (the conjectured optimal lower bound) for a certain class of closed curves that have an additional geometrical property.