Polynomials and asymptotic constants in a resurgent problem from 't Hooft

Abstract

In a recent study of the quantum theory of harmonic oscillators, Gerard 't Hooft proposed the following problem: given $G(z)={\sum }_{n=1}^{\infty }\sqrt{n}{z}^n$ for $|z|<1$, find its analytic continuation for $|z|\ge 1$, excluding a branch-cut $z\in [1,\infty )$. A solution is provided by the bilateral convergent sum $G(z)=\frac{1}{2}\sqrt{\pi }{\sum }_{n=-\infty }^{\infty }(2\pi \mathrm{i}n-\log (z){)}^{-3/2}$. On the negative real axis, $G(-{\mathrm{e}}^u)$ has a sign-constant asymptotic expansion in $1/u^2$, for large positive $u$. Optimal truncation leaves exponentially suppressed terms in an asymptotic expansion ${\mathrm{e}}^{-u}{\sum }_{k=0}^{\infty }{P}_k(x)/u^k$, with $P_0(x)=x-\frac{2}{3}$ and $P_k(x)$ of degree $2k+1$ evaluated at $x=u/2-\lfloor u/2\rfloor$. At large $k$, these polynomials become excellent approximations to sinusoids. The amplitude of $P_k(x)$ increases factorially with $k$ and its phase increases linearly, with $P_k(x)\sim \sin ((2k+1)C-2\pi x)R^{2k+1}\Gamma (k+\frac{1}{2})/\sqrt{2\pi }$, where $C\approx 1.0688539158679530121571$ and $R\approx 0.5181839789815558726739$ are asymptotic constants satisfying $R\exp (\mathrm{i}C)=\sqrt{-1/(2+\pi \mathrm{i})}$.

arXiv comment14 pages, 1 figure. Accepted for publication in Letters in Mathematical Physics. The exposition was improved based on the referees' comments

References in corpus (1)

• Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal