expansion $$\textbf{K}_{\cal H}(\vec{x},\vec{y};\beta)= \sum_{\ell=1}^{N_{\rm zm}} \Xi_{0\ell}(\vec{x}) \, \Xi_{0\ell}^\dagger (\vec{y}) + \sum_{n=1}^{N_B} \Xi_n(\vec{x}) \, \Xi_n^\dagger (\vec{y}) e^{-\beta \omega_n^2} + \int [dk_1 dk_2] \, \Xi_{\vec{k}}(\vec{x}) \, \Xi_{\vec{k}}^\dagger(\vec{y}) \, e^{-\beta \omega^2(\vert\vec{k}\vert)} \label{integralkernel01}$$ Here $N_{\rm zm}$ denotes the number of zero modes $\Xi_{0\ell}(\vec{x})$, $N_{\rm zm}$ linearly independent functions belonging to the algebraic kernel of ${\cal H}$, $N_B$ is the number of bound states $\Xi_n(\vec{x})$ in ${\rm Spec}({\cal H})$, and $\Xi_{\vec{k}}(\vec{x})$ are the continuous spectrum eigenfunctions of the operator ${\cal H}$. They are $D$-component functions and form an orthonormal basis in the Hilbert space $\oplus_{a=1}^D L^2_a(\mathbb{R}^2)$. The $\beta=0$ (infinite temperature) condition in (\[heatequation\]) is derived from the completeness of the set of ${\cal H}$-eigenfunctions.

The standard Gilkey-de Witt cunning strategy is based in using the knowledge of the ${\cal H}_0$ heat kernel. In a normalizing square of area $L^2$ it reads: $$\mathbf{K}_{{\cal H}_0} (\vec{x},\vec{y};\beta)= \frac{l^2}{4\pi \beta} \, e^{-\frac{\|\vec{x}-\vec{y}\|^2}{4\beta}} \, e^{-\beta \textbf{u}^2} \hspace{0.5cm}, \hspace{0.5cm} e^{-\beta \textbf{u}^2}= {\rm diag}\,\{e^{-\beta u_1^2}, \dots, e^{-\beta u_D^2}\} \, \, \, , \, \, \, l=mL\label{hk0}$$ and therefore in this context the ${\cal H}$ heat kernel is assumed to follow the factorization: $$\textbf{K}_{\cal H}(\vec{x},\vec{y};\beta)= \textbf{A}(\vec{x},\vec{y};\beta) \, \textbf{K}_{{\cal H}_0} ( \vec{x},\vec{y};\beta) \label{factorizacion0}$$ Plugging this ansatz into the heat equation (\[heatequation\]) another equation for 