New method to evaluate divergent series via the Wigner function

Renormalization-group Resummation of Divergent Series of the Perturbative Wave Function of Quantum Anharmonic Oscillator

The renormalization group method is applied for obtaining the asymptotic form of the wave function of the quantum anharmonic oscillator by resumming the perturbation series. It is shown that the resumed series is the cumulant of the naive perturbation series. Working out up to the sixth order and performing a further resummation proposed by Bender and Bettencourt, we find that the agreement with the WKB result becomes worse in the higher orders than the fourth at which the agreement is the best.

Convergent and Divergent Series in Physics

The aim of this review, based on a series of four lectures held at the 22nd "Saalburg" Summer School (2016), is to cover selected topics in the theory of perturbation series and their summation. The first part is devoted to strategies for accelerating the rate of convergence of convergent series, namely Richardson extrapolation, and the Shanks transformations, and also covers a few techniques for accelerating the convergence of Fourier series. The second part focuses on divergent series, and on the tools allowing one to retrieve information from them. These techniques include Euler summation, …

Generalized binomial transform applied to the divergent series

The divergent series for a function defined through Lapalce integral and the ground state energy of the quartic anharmonic oscillator to large orders are studied to test the generalized binomial transform which is the renamed version of $δ$-expansion proposed recently. We show that, by the use of the generalized binomial transform, the values of functions in the limit of zero of an argument is approximately computable from the series expansion around the infinity of the same argument. In the Laplace integral, we investigate the subject in detail with the aid of Mellin transform. In the anharmo…

Resummation of the Divergent Perturbation Series for a Hydrogen Atom in an Electric Field

We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescr…

A new method to sum divergent power series: educated match

We present a method to sum Borel- and Gevrey-summable asymptotic series by matching the series to be summed with a linear combination of asymptotic series of known functions that themselves are scaled versions of a single, appropriate, but otherwise unrestricted, function $Φ$. Both the scaling and linear coefficients are calculated from Padé approximants of a series transformed from the original series by $Φ$. We discuss in particular the case that $Φ$ is (essentially) a confluent hypergeometric function, which includes as special cases the standard Borel-Padé and Borel-Leroy-Padé methods. A p…

从发散级数到几何：量子度量的重现性

From Divergent Series to Geometry: Resurgence of the Quantum Metric

本研究针对非简谐振子的量子度量张量（QMT）微扰展开发散问题，通过高阶微扰理论验证其阶乘增长特性，结合重现理论与Borel–Padé重求和得到准确结果，将重现技术拓展至QMT领域，揭示量子几何与非微扰物理的关联。

Fast Summation of Divergent Series and Resurgent Transseries in Quantum Field Theories from Meijer-G Approximants

We demonstrate that a Meijer-G-function-based resummation approach can be successfully applied to approximate the Borel sum of divergent series, and thus to approximate the Borel-Écalle summation of resurgent transseries in quantum field theory (QFT). The proposed method is shown to vastly outperform the conventional Borel-Padé and Borel-Padé-Écalle summation methods. The resulting Meijer-G approximants are easily parameterized by means of a hypergeometric ansatz and can be thought of as a generalization to arbitrary order of the Borel-Hypergeometric method [Mera {\it et al.} Phys. Rev. Lett. …

Optical homodyne tomography with polynomial series expansion

We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner function and the marginal distribution and discretize Fourier space. We show that this technique solves most technical difficulties encountered with kernel deconvolution based methods and reconstructs overall better and smoother Wigner functions. We also give estimators of the reconstruction errors for both methods and show improvement in noise handling proper…

通过有限部分积分将Stieltjes级数延拓至大参数区域

Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration

本文提出利用Hadamard有限部分积分处理发散负幂矩，将Stieltjes积分的强渐近收敛展开用于对所有微扰区域求和发散级数，并成功计算了多种非谐振子与漏斗势的基态能量。

Wigner正量子态纯度上界的研究：验证Wigner熵猜想

Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture

本文针对Wigner熵猜想，通过构建下界序列和纯度条件，证明了特定μ值下Wigner非负态满足猜想，并指出纯度方法需额外约束以接近极值情况。

Hudson定理在SU(1,1)离散系列中失效

Hudson's theorem fails for the SU(1,1) discrete series

研究证明Hudson定理在SU(1,1)离散系列的弯曲相空间失效，Wigner正纯态集合大于相干轨道，并解析确定正性窗口受混合角限制。

Wigner function non-classicality as indicator of quantum chaos

We propose a Wigner function based parameter that can be used as an indicator of quantum chaos. This parameter is defined as "entropy" from the time-dependence of "non-classicallity" proposed in \cite{KZ04}. We perform our considerations for the system of damped nonlinear (Kerr-like) oscillator excited by a series of ultra-short external pulses.

Angular momentum decomposition of the three-dimensional Wigner harmonic oscillator

In the Wigner framework, one abandons the assumption that the usual canonical commutation relations are necessarily valid. Instead, the compatibility of Hamilton's equations and the Heisenberg equations are the starting point, and no further assumptions are made about how the position and momentum operators commute. Wigner quantization leads to new classes of solutions, and representations of Lie superalgebras are needed to describe them. For the n-dimensional Wigner harmonic oscillator, solutions are known in terms of the Lie superalgebras osp(1|2n) and gl(1|n). For n=3N, the question arises …

Some properties of Wigner $3j$ coefficients: non-trivial zeros and connections to hypergeometric functions

The contribution of Jacques Raynal to angular-momentum theory is highly valuable. In the present article, I intend to recall the main aspects of his work related to Wigner $3j$ symbols. It is well known that the latter can be expressed with a hypergeometric series. The polynomial zeros of the $3j$ coefficients were initially characterized by the number of terms of the series minus one, which is the degree of the coefficient. A detailed study of the zeros of the $3j$ coefficient with respect to the degree $n$ for $J = a + b + c \leq 240$ ($a$, $b$ and $c$ being the angular momenta in the first …

弱对抗神经推前方法用于威金方程

Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

本文提出弱对抗神经推前方法用于求解威金传输方程，通过结构观察将非局部伪微分势能算子与平面波测试函数积分转化为狄拉克δ函数，实现对威金势能核的精确反转，无需截断莫伊尔级数且能处理量子系统相空间动力学。

量子统计力学概率的幂级数及其在氦气第二维里系数中的结果

Power Series for the Quantum Statistical Mechanics Probability with Results for the Second Virial Coefficient of Helium

本文提出可递归生成项的Wigner-Kirkwood对易子幂级数，其计算的氦气第二维里系数与65K以上实验值吻合，并探讨了通用量子蒙特卡洛算法的前景。