encompassed in formula (\[generaloperator\]) for $D=4$ and the following assignments of vacuum diagonal matrix and first-order PDO vector field $$\textbf{v}={\rm diag}\,\{1,1,1,1\} \quad , \quad \quad
\textbf{Q}_k(\vec{x})= \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2V_k \\ 0 & 0 & 2V_k & 0 \end{array} \right)\, \, ,$$ whereas the $4\times 4$-matrix potential well is defined in (\[vspot\]).

The Gilkey-de Witt approach aims to construct a power series expansion of the ${\cal H}$-spectral heat trace $h_{\cal H}(\beta) = {\rm Tr}_{L^2} \, e^{-\beta \, {\cal H}}$ by taking advantage of the fact that this function can be obtained from integration all over the plane of the diagonal ${\cal H}$-heat equation kernel: $$h_{{\cal H}}(\beta) = \int_{\mathbb{R}^2}\, d^2 x \,\, {\rm tr}\, \textbf{K}_{\cal H} (\vec{x},\vec{x};\beta) \label{heatfunction0}$$ i.e., the trace in both the $L^2$-functional and $D\times D$-matrix senses of the integral kernel of the ${\cal H}$-heat equation: $$\Big( \frac{\partial}{\partial \beta} + {\cal H} \Big) \mathbf{K}_{\cal H} (\vec{x},\vec{y},\beta) =0 \hspace{0.5cm} , \hspace{0.5cm} \mathbf{K}_{\cal H}(\vec{x},\vec{y};0)=\delta^{(2)}(\vec{x}-\vec{y} ) \, \textbf{I}_{D\times D} \, \, . \label{heatequation}$$ Completeness of the eigenfunctions of ${\cal H}$ allows to write the fundamental solution of equation (\[heatequation\]) as the 