Asymptotics for Toeplitz operators with symbol an indicator function

We prove an off-diagonal expansion of the kernel of the Toeplitz operator whose symbol is the indicator function of a compact domain with smooth boundary in a complete symplectic manifold of bounded geometry. Using our approach, we extend two results to the non-compact setting: the first concerns the asymptotics of the trace of polynomials in this operator, and the second establishes a Weyl law for this Toeplitz operator.