This paper provides a criterion of the Watson-Nevanlinna type for the ”logarithmic Borel summability”, by which a formal power series \(\sum a_ kz^ k\) admits a ”logarithmic Borel sum” g(z) if (a) \(B(t)=(\alpha /\pi)^{1/2}\sum^{\infty}_{k=0}a_ ke^{-k^ 2/4\alpha}t^ k\) converges for small t; (b) it has an analytic continuation to some neighborhood of \({\mathbb{R}}_+\); (c) \(g(z)=\int^{\infty}_{0}\exp (- (\log (t/z))^ 2)B(t)t^{-1}dt\) is an absolutely convergent integral for some \(z\neq 0\). Explicitly, let f(z) be analytic on the Riemann surface of log z for \(-\infty <Re(\log z)<c_ 0\) (c\(\in {\mathbb{R}})\) and let it satisfy the estimate \[ (1)\quad | R_ N(z)| \equiv | f(z)-\sum^{N-1}_{k=0}a_ kz^ k| \leq A\delta^ Ne^{N^ 2/4\alpha}| z|^ N\exp (\alpha \theta^ 2-2\alpha \phi_ 0| \theta |) \] for constants A, \(\delta\), \(\alpha\), \(\phi_ 0\) independent of \(| z|\), of \(\theta =\arg (z)\) and of \(N\in {\mathbb{N}}\). Then the series (a), which is convergent for small t, defines a function B(t) analytic in \(S(\delta,\phi_ 0)=\{t/| t| <\delta^{-1}\) or \(| \arg (t)| <\phi_ 0\}\) such that \[ (2)\quad | B^{(N)}(t)| \leq \]
\[ \leq A^ N_ 1e^{\alpha (\log t)^ 2}e^{-2\alpha c(\log t)}t^{-N}\sum^{N}_{r=0}\left( \begin{matrix} N\\ r\end{matrix} \right)(N-r)!\sum^{r}_{j=0}r!(j!)^{-1}| \log t|^ j \] for \(t>0\), for any fixed c \((-\infty,c_ 0)\). Besides, f(z) admits the integral representation (c) for \(-\infty <Re(\log z)<c_ 0\). Conversely, if B(t) is analytic in \(S(\delta,\phi_ 0)\) and satisfies (2), then the integral (c) defines a function f(z) analytic for \(-\infty <Re(\log z)<c_ 0\), satisfying an estimate of the type (1).
The applicability to the eigenvalues E(z) of the exponential anharmonic oscillator \(H(z)=p^ 2+x^ 2+z e^ x\) is conjectured: in fact it is proved that each E(z) satisfies analyticity and remainder estimates of the type (1) in any fixed angular sector of the Riemann surface of log z, for small \(| z|\). On the other hand, the first approximant by the Trotter formula of \(Tr(e^{-tH(z)})\), that is \((2\pi t)^{- 1/2}\int^{+\infty}_{-\infty}e^{-t(x^ 2+ze^ x)}dx\), has logarithmically Borel summable expansion in powers of z.
The authors have also proved a criterion for a ”generalized logarithmic Borel summability” [J. Math. Phys. 25, 3439-3443 (1984)] for power series with divergence of the type \(| a_ k| \sim (pk)!\exp (\alpha k^ 2/4)\) as \(k\to \infty\), \(p\in {\mathbb{N}}\).