Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori

Let $P$ be a Laplace type operator acting on a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold given locally by $P= - [g^{\mu\nu} u(x)\partial_\mu\partial_\nu + v^\nu(x)\partial_\nu + w(x)]$ where $u,\,v^\nu,\,w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr} (a e^{-tP}) \underset{t \downarrow 0^+}{\sim} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written locally as $a_r(a, P)(x) = \text{tr}[a(x) \mathcal{R}_r(x)]$. The computation of $\mathcal{R}_2$ is performed opening the opportunity to calculate the modular scalar curvature for noncommutative tori.