Curvature induced quantum potential on deformed surfaces

TL;DR

This paper explores how surface curvature influences quantum particles, deriving quantum potentials for Gaussian bumps, solving inverse geometric problems, and identifying surfaces that enable particle binding based on angular momentum.

Contribution

It introduces a method to derive curvature-induced quantum potentials and solves the inverse problem for rotational surfaces, revealing new particle binding scenarios.

Findings

Gaussian bump creates bound states for zero angular momentum particles

Inverse problem solutions for rotational surfaces with prescribed quantum potentials

Existence of rotational surfaces allowing binding of particles with arbitrary angular momentum

Abstract

We investigate the effect of curvature on the behaviour of a quantum particle bound to move on a surface. For the Gaussian bump we derive and discuss the quantum potential which results in the appearance of a bound state for particles with vanishing angular momentum. The Gaussian bump provides a characteristic length for the problem. For completeness we propose an inverse problem in differential geometry, i.e. what deformed surfaces produce prescribed curvature induced quantum potentials. We solve this inverse problem in the case of rotational surfaces. We also show that there exist rotational surfaces in the form of a circular strip around the axis of symmetry which allow particles with generic angular momentum to bind.