On Perturbed Two-Dimensional Dirac Operators

In this paper, we study a perturbed two-dimensional Dirac operator D, which solves D2=Δ2+u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^2=\Delta _2+u$$\end{document} with non-vanishing potential u=u(x,y)≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u=u(x,y)\ne 0$$\end{document}. By introducing the so-called formal operators, we prove that a regular and normal solution D exists, if and only if u is splitting, and that is u(x,y)=f(x)+g(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(x,y)=f(x)+g(y)$$\end{document}.