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 $P=-[g^{\mu \nu }u(x){\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }+v^{\nu }(x){\mathrm{\partial }}_{\nu }+w(x)]$ where $u,v^{\nu },w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. For any $a\in \mathrm{\Gamma }(\text{End}(V))$, 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}$wherethecoefficients$a_r(a, P)$canbewrittenlocallyas$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.