Singularity Structure and Stability Analysis of the Dirac Equation on the Boundary of the Nutku Helicoid Solution

TL;DR

This paper investigates the singularity and stability of the Dirac equation on the boundary of the Nutku helicoid space, revealing higher singularities and loss of integrals, which hinder solutions in known functions.

Contribution

It provides a detailed analysis of the boundary Dirac equation's singularity structure and stability, highlighting differences from bulk solutions and introducing new examples involving Heun equations.

Findings

Boundary Dirac equation has higher singularity than bulk solutions.

Loss of an independent integral of motion on the boundary.

Stability analysis of helicoid and catenoid cases.

Abstract

Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heun's equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example where one encounters a form of the Heun equation.