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Use the formal pseudodifferential composition rule on each coefficient $w_i(x)$ and then the exponential eigenfunction convention. For every integer $n$,ConsequentlyCollecting all terms of a fixed power of the spectral parameter gives the useful answerThe sum defining each $v_s$ is finite. For example,andThis is the action on a formal pseudodifferential wave function. It is a formal expansion near $k=\infty$, with $k^{\beta }$ carried as a formal prefactor. It does not assert convergence for finite $k$, and negative powers are not evaluated at $k=0$.