Renormalization-Group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

Kunihiro, Teiji

doi:10.1143/ptps.131.459

Cited by 21 publications

(37 citation statements)

References 13 publications

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“…Fundamental analysis of the RG condition, RG 3 , and the RG method is presented in [20][21][22][23]10]. First, it was shown that RG 3 is an envelope equation in the sense of classical differential geometry.…”

Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

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For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O( 2 ), where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ ).

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Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…This condition is naturally satisfied if we restrict that t 0 < t < t ′ 0 (or t ′ 0 < t < t 0 ) because the limit t ′ 0 → t 0 is to be taken. Thus when a perturbative expansion is used for constructing X(t; t 0 , W (t 0 )), we demand a more restrictive RG equation as given by [27] …”

Section: Perturbative Rg Equationmentioning

confidence: 99%

“…We remark that a geometrical interpretation of this equation has been given on the basis of the classical theory of envelopes [25,26,27]: When t 0 is varied, X(t; t 0 , W (t 0 )) gives a family of curves with t 0 being a parameter characterizing curves. Then Eq.…”

Section: Perturbative Rg Equationmentioning

confidence: 99%

“…The resultant forms of the unperturbed solutions composed of the secular terms which vanish at t = t 0 and the functions independent of the unperturbed solution. The final forms turn out to be the same as those given in the scheme adopted in [25,26,27], where unstable manifolds and cases with a Jordan cell were not dealt with though. The resultant initial value represent the invariant manifold, and the RG equation gives the slow dynamics on the manifold.…”

Section: Simple Examplesmentioning

confidence: 99%

“…He also developed a short-cut prescription for the renormalization procedure without introducing an intermediate time τ but utilizing an arbitrary initial time t 0 . In latest papers [27], it was shown that the RG method can be well formulated as the method dealing with the initial values at arbitrary t = t 0 ; the initial values at t = t 0 are determined so that the unperturbative solutions which are valid only locally around t = t 0 are continued smoothly; this procedure is nothing but to construct the envelope of the perturbative solutions. He emphasized that the RG method is a powerful tool for reduction of evolution equations and demonstrated it by applying the method to obtain so-called the amplitude equations for systems of equations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

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145

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The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t 0 = t is naturally understood where t 0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

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Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

Hatta

Kunihiro

2002

Annals of Physics

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The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t 0 to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t 0 , and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method. C 2002 Elsevier Science (USA)

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Renormalization-Group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

Cited by 21 publications

References 13 publications

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

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