Heat trace asymptotics and compactness of isospectral potentials for the Dirichlet Laplacian

Let $\Omega$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overline{\Omega};\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(\Omega)$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V \in C_0^\infty (\Omega)$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t \downarrow 0$ for any matrix ${\bf a}$ whose coefficients are homogeneous of degree $0$. In the particular case where $A$ is the Dirichlet Laplacian in $\Omega$, that is when ${\bf a}$ is the identity of $\R^{n^2}$, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that $L^\infty$-bounded sets of isospectral potentials of $A$ are $H^s$-compact for $s <2$.