Variational Resummation of Divergent Series with known Large-Order Behavior

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. …

带共形映射的Borel重求和法与四次非谐振子的基态能量

Borel Resummation Method with Conformal Mapping and the Ground State Energy of the Quartic Anharmonic Oscillator

本文研究一维四次非谐振子基态能量的发散微扰展开重求和问题，提出将Borel-Pade方法与Borel平面共形映射相结合，并计算前50个修正项系数，证明该方法在强耦合区域显著优于标准Borel重求和和对角Pade近似。

Scaling property of variational perturbation expansion for general anharmonic oscillator

We prove a powerful scaling property for the extremality condition in the recently developed variational perturbation theory which converts divergent perturbation expansions into exponentially fast convergent ones. The proof is given for the energy eigenvalues of an anharmonic oscillator with an arbitrary $x^p$-potential. The scaling property greatly increases the accuracy of the results.

Variational versus perturbative relativistic energies for small and light atomic and molecular systems

Variational and perturbative relativistic energies are computed and compared for two-electron atoms and molecules with low nuclear charge numbers. In general, good agreement of the two approaches is observed. Remaining deviations can be attributed to higher-order relativistic, also called non-radiative quantum electrodynamics (QED), corrections of the perturbative approach that are automatically included in the variational solution of the no-pair Dirac$-$Coulomb$-$Breit (DCB) equation to all orders of the $α$ fine-structure constant. The analysis of the polynomial $α$ dependence of the DCB ene…

Excited states from eigenvector continuation: the anharmonic oscillator

Eigenvector continuation (EC) has recently attracted a lot attention in nuclear structure and reactions as a variational resummation tool for many-body expansions. While previous applications focused on ground-state energies, excited states can be accessed on equal footing. This work is dedicated to a detailed understanding of the emergence of excited states from the eigenvector continuation approach. For numerical applications the one-dimensional quartic anharmonic oscillator is investigated, which represents a strongly non-perturbative quantum system where the use of standard perturbation te…

Perturbation theory without power series: iterative construction of non-analytic operator spectra

It is well known that quantum-mechanical perturbation theory often give rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place. Using the elementary technique of relaxed fixed-point iteration, I obtain convergent expressions for various challenging ground states wavefunctions, including quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing Rayleigh-Schrödinger coefficients), all at arbitarily strong coupling. Th…

Rescaled Perturbation Theory

A non-perturbative method which can go beyond the weak coupling perturbation theory is introduced. Essential idea is to formulate a set of exact differential equations as a function of the coupling strength $g$. Unlike other resummation in which information of the higher order terms is necessary, we only need a leading order perturbative formula in every step to reach the large value of $g$. The method is applied to the quantum anharmonic oscillator and quantum double well potential in one dimension. Both are known to have divergent series in the weak coupling perturbation and the latter is no…

Determination of the exchange interaction energy from the polarization expansion of the wave function

The exchange contribution to the energy of the hydrogen atom interacting with a proton is calculated from the polarization expansion of the wave function using the conventional surface-integral formula and two formulas involving volume integrals: the formula of the symmetry-adapted perturbation theory (SAPT) and the variational formula recommended by us. At large internuclear distances $R$, all three formulas yield the correct expression $-(2/e)Re^{-R}$, but approximate it with very different convergence rates. In the case of the SAPT formula, the convergence is geometric with the error fallin…