Energy Spectrum of a constrained Quantum Particle and the Willmore Energy of the constraining Surface

TL;DR

This paper derives bounds on the energy gap of a quantum particle constrained to a surface in 3D space and identifies geometric conditions for stationary, uniformly distributed wave functions.

Contribution

It introduces new geometric bounds for the energy spectrum and characterizes conditions for uniform wave functions on constrained surfaces.

Findings

Bounds on the first energy level gap are established.

Conditions for stationary and uniform wave functions are identified.

Connections between geometric properties and quantum states are demonstrated.

Abstract

Geometric and topological bounds are obtained for the first energy level gap of a particle constrained to move on a compact surface in 3-space. Moreover, geometric properties are found which allows for stationary and uniformly distributed wave functions to exist on the surface.