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

TL;DR

This paper establishes a lower bound of 0.6085 for the ground state energy of a quantum particle on a closed curve with curvature-based potential, and proves a conjectured bound of 1 for certain geometrically constrained curves.

Contribution

It provides the first rigorous lower bounds for the ground state energy in this geometric quantum setting, including a proof of the conjectured optimal bound for specific curves.

Findings

Ground state energy cannot be lower than 0.6085 for the given system.

For certain curves with additional properties, the energy cannot be lower than 1.

The results support the conjecture of the optimal lower bound being 1.

Abstract

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