Let $\operatorname{Op}_h(x,D)(a)$ denote the Weyl-quantisation of a symbol $a$. Is there an explicit way to invert this pseudo-differential operator in an asymptotic series? By this I mean, can we find $(x,\xi) \mapsto \sum_{n=0}^{N} c_n(x,\xi)h^n$ such that $$a \sharp \left(\sum_{n=0}^{N} c_n h^n \right)=1 + O(h^{(n+1)})$$ for any $N \in \mathbb{N}$ explicitly?

The only thing that I can see is how to do this step by step. For example, the zero-th order term is given by taking $c_0:=a^{-1}$. Then, I would have to look how to subpress the next higher-order term and so on. But is there also a closed way to write down the inverse expansion?

If anything is unclear, please let me know.