Paramagnetism for Nonrelativistic Electrons and Euclidean Massless Dirac Particles

Abstract

We construct the manifold of zero-energy eigenstates for a nonrelativistic spin-\textonehalf{} particle moving in a plane in an external magnetic field $B(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})=\ensuremath{\Sigma}{n=0}^{N}{\ensuremath{\lambda}}_{n}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{c}}}_{n})$, with ${{\ensuremath{\lambda}}_{n}}$ and ${{\stackrel{\ensuremath{\rightarrow}}{\mathrm{c}}}_{n}}$ arbitrary reals and ${{k}_{n}}$ positive integers. For a given $B$ the ground state is infinitely degenerate and the manifold of eigenfunctions is parametrized by a point in ${R}^{2(2{k}_{max}+1)}$. For such $B$'s we prove paramagnetism with arbitrary external potential $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$.