Heat trace asymptotics and boundedness in the second order Sobolev space of isospectral potentials for the Dirichlet Laplacian

Let Ω be a C^∞-smooth bounded domain of Rⁿ, n≥1, and let the matrix a∈C^∞(Ω¯;Rⁿ²) be symmetric and uniformly elliptic. We consider the L²(Ω)-realization A of the operator −div (a∇·) with Dirichlet boundary conditions. We perturb A by some real valued potential V∈C₀^∞(Ω) and note A_V=A+V. We compute the asymptotic expansion of tr (e^−tA_V−e^−tA) as t↓0 for any matrix a with constant coefficients. In the particular case where A is the Dirichlet Laplacian in Ω, that is when a is the identity of Rⁿ², we make the four main terms appearing in the asymptotic expansion formula explicit and prove that L^∞-bounded sets of isospectral potentials of A are bounded in H²(Ω).