Paramagnetism for Nonrelativistic Electrons and Euclidean Massless Dirac Particles

We construct the manifold of zero-energy eigenstates for a nonrelativistic spin-½ particle moving in a plane in an external magnetic field $B\left(\overrightarrow{\mathrm{x}}\right)=\Sigma {n=0\mathrm{}}^N{\lambda }_n(\overrightarrow{\mathrm{x}}-{\overrightarrow{\mathrm{c}}}_n)$, with $\left\{{\lambda }_n\right\}$ and $\left\{{\overrightarrow{\mathrm{c}}}_n\right\}$ arbitrary reals and $\left\{k_n\right\}$ positive integers. For a given $B$ the ground state is infinitely degenerate and the manifold of eigenfunctions is parametrized by a point in $R^{2(2k_{max}+1)}$. For such $B$'s we prove paramagnetism with arbitrary external potential $V\left(\overrightarrow{\mathrm{x}}\right)$.