\quad \quad \lim_{\beta\to 0} \textbf{G}(\beta)=0 \qquad \quad , \qquad \quad \lim_{\beta\to 0}\textbf{C}(\vec{x},\vec{y};\beta)= \textbf{I}_{D\times D} \label{initialconditionC}$$ The matrix density $\textbf{C}(\vec{x},\vec{y};\beta)$, like $\textbf{A}(\vec{x},\vec{y};\beta)$ in the standard GdW method, relates the positive part of ${\rm Spec}{\cal H}$ to ${\rm Spec}{\cal H}_0$ in the ${\cal H}$-heat kernel, whereas the second term in the right hand side of (\[factorizacion1\]) encodes the contribution of zero modes.

The power series expansion $$\textbf{C}(\vec{x},\vec{y};\beta)=\sum_{n=0}^\infty \textbf{c}_n(\vec{x},\vec{y}) \, \beta^n \label{expansion5}$$ together with the factorization (\[factorizacion1\]) is plugged into the heat equation (\[heatequation\]) as in the standard GdW procedure. The PDE (\[heatequation\]) is converted thereafter into the following relations between the coefficients of the modified GdW expansion and their derivatives: $$\begin{aligned}
&& -\,\frac{1}{2\beta} \,(\vec{x}-\vec{y}) \cdot \vec{\textbf{Q}}(\vec{x}) \,\, \textbf{c}_0(\vec{x},\vec{y}) + \sum_{n=0}^\infty \Big[(n+1)\textbf{c}_{n+1}(\vec{x},\vec{y}) - \Delta \textbf{c}_n(\vec{x},\vec{y})+ (\vec{x}-\vec{y})\cdot \vec{\nabla} \textbf{c}_{n+1}(\vec{x},\vec{y}) + \nonumber \\ && + \, \textbf{U}(\vec{x})\, \textbf{c}_n(\vec{x},\vec{y}) + [\textbf{u}^2,\textbf{c}_n(\vec{x},\vec{y})] + \vec{\textbf{Q}}(\vec{x}) \cdot \vec{\nabla} \textbf{c}_n(\vec{x},\vec{y}) - \frac{1}{2} \,(\vec{x}-\vec{y})\cdot \vec{\textbf{Q}}(\vec{x}) \,\,\textbf{c}_{n+1}(\vec{x},\vec{y}) \Big] \beta^n + \nonumber \\ && +\, \sum_{\ell=1}^{N_{\rm zm}} 4 \pi \Big[ \Xi_{0\ell}(\vec{x}) \, \Xi_{0\ell}^\dagger (\vec{y}) \Big( \beta \frac{d\textbf{G}}{d\beta}(\beta) + \textbf{G}(\beta) \Big) + (\vec{x}-\vec{y})\cdot \vec{\nabla} \Xi_{0\ell}(\vec{x}) \,\, \Xi_{0\ell}^\dagger (\vec{y}) \, \textbf{G}(\beta) - \nonumber\\ && -\,\frac{1}{2}\, (\vec{x}-\vec{y})\cdot \vec{\textbf{Q}}(\vec{x}) \,\, \Xi_{0\ell}(\vec{x}) \,\, \Xi_{0\ell}^\dagger (\vec{y}) \, \textbf{G} (\beta) \Big] e^{\beta \textbf{u}^2} =\textbf{0} \, . \label{mgdwrr}\end{aligned}$$ Taking into account 