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On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: using the holomorphic functional calculus one has (under suitable conditions) $$ A^{-s} = \frac{i}{2 \pi} \int_\Gamma \,\lambda^{-s}(A - \lambda)^{-1} \,d\lambda $$ with $\Gamma$ a contour surrounding the spectrum of $A$. Essentially many properties of this operator can be deduced from the asymptotic expansion for the local resolvent symbol $$ \sigma_{(A - \lambda)^{-1}}(x,\xi,\lambda) \backsim \sum_{j \ge 0} b_{ - m - j}(x,\xi,\lambda) \quad (\text{as }(\xi,\lambda^{1/m}) \to \infty) $$ and there is an explicit recursive formula for the terms $b_{ - m - j}$, given by \begin{align} b_{-m} &= (a_m - \lambda)^{-1} \\ b_{ - m -j} &= - b_{-m}\sum_{\substack{l < j \\ l + k + |\alpha| = j}} \frac{1}{\alpha!}(\partial^\alpha_\xi b_{ - m - l})D^\alpha_x a_{m - k} \qquad (\text{for } j > 0) \end{align} where $a \backsim \sum_{i \ge 0} a_{m - i}$ is an asymptotic expansion for the local symbol $a$ of $A$.

I was wondering whether it is known if the same recursive formula for the resolvent symbols (the $b_{ - m - j}$) holds if one works on a noncompact manifold instead, I would use it formally for a start but eventually I would like to understand this better. References to papers specifically dealing with this issue would be very helpful, my initial search via Google was not very successful (I found two papers that dealt with complex powers on noncompact manifolds from a more abstract point of view) but I may be looking in the wrong place.

Edit: Essentially I think that the formula should still hold provided that one has a parameter - dependent calculus for operators (so that the composition formula for symbols holds), perhaps I should be looking for this instead.

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