continuous spectrum.

It is interesting at this point to summarize a heat function proof of the Atiyah-Singer-Weinberg index theorem [@Weinberg1979:prd; @Weinberg2012]. Weinberg showed the existence of $2N$ linearly independent zero modes $\xi_0(\vec{x})$ of ${\cal H}^+$ (eigenfunctions with zero eigenvalues). Weinberg’s proof rely on a supersymmetric structure built on perturbations of solutions of (\[pde1\]) which are still solutions. Perturbing the PDE (\[pde1\]) system of three equations together with the background gauge one finds that new solutions arise, complying with the background gauge, if and only if the perturbations belong to the kernel of the deformation operator ${\cal D}$: $$\hspace{-2cm}{\cal D}= \left( \begin{array}{cccc}
-\partial_2 & \partial_1 & \psi_1 & \psi_2 \\
-\partial_1 & -\partial_2 & -\psi_2 & \psi_1 \\
\psi_1 & -\psi_2 & -\partial_2 + V_1 & -\partial_1 -V_2 \\
\psi_2 & \psi_1 & \partial+V_2 & -\partial_2 + V_1
\end{array} \right) \quad ,\quad \quad \hspace{1cm}{\cal D}\xi_0(\vec{x})=0 \label{zeromode1} \, \, .$$ It is easy to check that the Hessian ${\cal H}^+$ factorizes as the product of the ${\cal D}$ operator times its adjoint: ${\cal H}^+={\cal D}^\dagger \, {\cal D}$. Besides of showing that the four-vector columns $\xi_0(\vec{x})$ are zero modes of ${\cal H}^+$, this factorization hides a supersymmetric quantum mechanical structure where the 