Obtain an explicit analytic expression for the WKB amplitude correction

Determine whether the first-order WKB correction $\mathcal{F}_1$ for the Starobinsky Wheeler-DeWitt equation admits an explicit analytic expression after the leading function $\mathcal{F}_0$ has been obtained as an expansion in $x=e^{-\alpha\phi}$.

Background

The Starobinsky calculation constructs the leading WKB function F0\mathcal{F}_0 iteratively in powers of x=e−αϕx=e^{-\alpha\phi} and states that the expansion can be continued to arbitrary order. The next WKB function F1\mathcal{F}_1 satisfies a more difficult equation, and the thesis leaves unresolved whether that equation yields an explicit closed expression, despite allowing analytic calculation order by order in xx.

References

Similarly, since in theory we can get a full explicit expression for $\mathcal{F}{0}$, we may then plug it into (\ref{second}) to obtain $\mathcal{F}{1}$. It is not clear whether we would be able get an explicit expression for it from that, but overall we can calculate it analytically up to an arbitrary order of $x$.

— A quantum probabilistic approach to the emergence of an inflationary Starobinsky Universe  (2610.19616 - Kaimakkamis, 14 Oct 2026) in Chapter 5, subsection "Explicit calculation for the Starobinsky model"