Transformation of the asymptotic perturbation expansion for the anharmonic oscillator into a convergent expansion

2004 ◽  
Vol 322 (3-4) ◽  
pp. 194-204 ◽  
Author(s):  
I.A. Ivanov
Keyword(s):  
Anharmonic Oscillator ◽  
Perturbation Expansion ◽  
Convergent Expansion ◽  
Asymptotic Perturbation

scholarly journals Asymptotic Power Series of Field Correlators

10.14311/1199 ◽  
2010 ◽  
Vol 50 (3) ◽  
Author(s):  
I. Caprini ◽  
J. Fischer ◽  
I. Vrkoč
Keyword(s):  
Power Series ◽  
Large Class ◽  
Complex Plane ◽  
Perturbation Expansion ◽  
Asymptotic Power ◽  
General Contour ◽  
Asymptotic Perturbation ◽  
Class Of Functions ◽  
Borel Type

We address the problem of ambiguity of a function determined by an asymptotic perturbation expansion. Using a modified form of theWatson lemma recently proved elsewhere, we discuss a large class of functions determined by the same asymptotic power expansion and represented by various forms of integrals of the Laplace-Borel type along a general contour in the Borel complex plane. Some remarks on possible applications in QCD are made.

scholarly journals Operator thermalisation in d > 2: Huygens or resurgence

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Julius Engelsöy ◽  
Jorge Larana-Aragon ◽  
Bo Sundborg ◽  
Nico Wintergerst
Keyword(s):  
Wave Equations ◽  
Free Field ◽  
Perturbation Expansion ◽  
Technical Modifications ◽  
Exponential Relaxation ◽  
Huygens Principle ◽  
Free Fields ◽  
Asymptotic Perturbation ◽  
Perturbative Corrections ◽  
Odd Dimensions

Abstract Correlation functions of most composite operators decay exponentially with time at non-zero temperature, even in free field theories. This insight was recently codified in an OTH (operator thermalisation hypothesis). We reconsider an early example, with large N free fields subjected to a singlet constraint. This study in dimensions d > 2 motivates technical modifications of the original OTH to allow for generalised free fields. Furthermore, Huygens’ principle, valid for wave equations only in even dimensions, leads to differences in thermalisation. It works straightforwardly when Huygens’ principle applies, but thermalisation is more elusive if it does not apply. Instead, in odd dimensions we find a link to resurgence theory by noting that exponential relaxation is analogous to non- perturbative corrections to an asymptotic perturbation expansion. Without applying the power of resurgence technology we still find support for thermalisation in odd dimensions, although these arguments are incomplete.

scholarly journals Modified Laplace Transformation Method and ITS Application to the Anharmonic Oscillator

1997 ◽  
Vol 12 (31) ◽  
pp. 5687-5709 ◽  
Author(s):  
Naoki Mizutani ◽  
Hirofumi Yamada
Keyword(s):  
Laplace Transformation ◽  
Anharmonic Oscillator ◽  
Perturbation Expansion ◽  
Exact Result ◽  
Transformation Method ◽  
Perturbation Series ◽  
Gaussian Integral ◽  
Non Gaussian ◽  
Laplace Integral Representation ◽  
Laplace Transformation Method

We apply a recently proposed approximation method to the evaluation of non-Gaussian integral and anharmonic oscillator. The method makes use of the truncated perturbation series by recasting it via the modified Laplace integral representation. The modification of the Laplace transformation is such that the upper limit of integration is cutoff and an extra term is added for the compensation. For the non-Gaussian integral, we find that the perturbation series can give an accurate result and the obtained approximation converges to the exact result in the N → ∞ limit (N denotes the order of perturbation expansion). In the case of the anharmonic oscillator, we show that several order result yields good approximation of the ground state energy over the entire parameter space. The large order aspect is also investigated for the anharmonic oscillator.

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