$\textbf{A}(\vec{x},\vec{y};\beta)$ (usually called transfer equation) arises that is solved by expanding $\textbf{A}$ as a power series in $\beta$.

This procedure is well behaved if the spectrum of the operator ${\cal H}$ is strictly positive provided that the infinite temperature condition $\textbf{A}(\vec{x},\vec{y};0)=\textbf{I}_{D\times D}$ is fixed. However, if the operator exhibits zero modes the factorization (\[factorizacion0\]) is inconsistent because the left and right members in (\[factorizacion0\]) behave in different ways at zero temperature, see (\[integralkernel01\]) and (\[hk0\]): $$\lim_{\beta \rightarrow +\infty} \textbf{K}_{{\cal H}} (\vec{x},\vec{y}; \beta) = \sum_{\ell=1}^{N_{zm}} \Xi_{0\ell}(\vec{x}) \, \Xi_{0\ell}^\dagger (\vec{y}) \quad , \qquad \quad \lim_{\beta\rightarrow +\infty} \mathbf{K}_{{\cal H}_0} (\vec{x},\vec{y};\beta)=0 \label{Kinfinity} \, \, ,$$ due to the fact that $\textbf{A}(\vec{x},\vec{y};\beta)$ grows as a power of $\beta$ when $\beta\to +\infty$. In order to amend this discrepancy we replace the factorization (\[factorizacion0\]) by the following one: $$\textbf{K}_{{\cal H}}(\vec{x},\vec{y};\beta)= \textbf{C}(\vec{x},\vec{y};\beta) \, \mathbf{K}_{{\cal H}_0} ( \vec{x},\vec{y};\beta) + \sum_{\ell=1}^{N_{\rm zm}} e^{-\frac{\|\vec{x}-\vec{y}\|^2}{4\beta}} \, \Xi_{0\ell}(\vec{x}) \, \Xi_{0\ell}^\dagger (\vec{y}) \,\, \textbf{G}(\beta) \, \, .
\label{factorizacion1}$$ Good agreement between the zero temperature regime when zero modes are present, together the usual conditions at infinity temperature not affected by zero modes, are guaranteed provided that the matrix function $\textbf{G}(\beta)$ and the matrix density $\textbf{C}(\vec{x},\vec{y};\beta)$ satisfy: $$\lim_{\beta \rightarrow +\infty} \textbf{G}(\beta)= \textbf{I}_{D\times D} \quad , 