标题： 多项式和渐近常数在't Hooft的突发问题中的应用 显示英文标题

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

摘要： 在对谐振子量子理论的最近研究中，Gerard 't Hooft提出了以下问题：给定$G(z)=\sum_{n=1}^\infty\sqrt{n}\,z^n$对于$|z|<1$，找到其在$|z|\ge1$的解析延拓，排除分支切割$z\in[1,\,\infty)$。由双边收敛和$G(z)=\frac12\sqrt{\pi}\sum_{n=-\infty}^\infty(2\pi{\rm i}n-\log(z))^{-3/2}$提供了解答。 在负实轴上，$G(-{\rm e}^u)$在$1/u^2$中具有一个符号不变的渐近展开式，当正的$u$很大时。 最优截断在渐近展开式中留下指数级抑制项${\rm e}^{-u}\sum_{k=0}^\infty P_k(x)/u^k$，其中$P_0(x)=x-\frac23$和$P_k(x)$在$2k+1$次多项式在$x=u/2-\lfloor u/2\rfloor$处的值。 在大$k$时，这些多项式成为正弦波的良好近似。 The amplitude of $P_k(x)$ increases factorially with $k$ and its phase increases linearly, with $P_k(x)\sim\sin((2k+1)C-2\pi x)R^{2k+1}\Gamma(k+\frac12)/\sqrt{2\pi}$, where $C\approx1.0688539158679530121571$ and $R\approx0.5181839789815558726739$ are asymptotic constants satisfying $R\exp({\rm i}\,C)=\sqrt{-1/(2+\pi{\rm i})}$. 显示英文摘要

摘要： 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 $|z|\ge1$, excluding a branch-cut $z\in[1,\,\infty)$. A solution is provided by the bilateral convergent sum $G(z)=\frac12\sqrt{\pi}\sum_{n=-\infty}^\infty(2\pi{\rm i}n-\log(z))^{-3/2}$. On the negative real axis, $G(-{\rm e}^u)$ has a sign-constant asymptotic expansion in $1/u^2$, for large positive $u$. Optimal truncation leaves exponentially suppressed terms in an asymptotic expansion ${\rm e}^{-u}\sum_{k=0}^\infty P_k(x)/u^k$, with $P_0(x)=x-\frac23$ 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 $P_k(x)\sim\sin((2k+1)C-2\pi x)R^{2k+1}\Gamma(k+\frac12)/\sqrt{2\pi}$, where $C\approx1.0688539158679530121571$ and $R\approx0.5181839789815558726739$ are asymptotic constants satisfying $R\exp({\rm i}\,C)=\sqrt{-1/(2+\pi{\rm i})}$.