A Recursion Technique for Deriving Renormalized Perturbation Expansions for One-Dimensional Anharmonic Oscillator

Dobrovolska, I. V.; Tutik, R. S.

doi:10.1142/s0217751x0100372x

Cited by 9 publications

(9 citation statements)

References 32 publications

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“…A great number of papers is devoted to the one-dimensional anharmonic oscillator, as in Sec. 2.7, considering its eigenvalues [2,[15][16][17][18][19][20][21][22][23][24][25][65][66][67][68] and wave functions [69].…”

Section: Linear Hamiltoniansmentioning

confidence: 99%

Interplay between Approximation Theory and Renormalization Group

Yukalov

2019

Phys. Part. Nuclei

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The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory in powers of these parameters, and even when small parameters exist, the related perturbative series are strongly divergent. Second, such perturbative series in powers of these parameters are rather short, so that the standard resummation techniques either yield bad approximations or are not applicable at all. The emphasis in the review is on the methods advanced and developed by the author. One of the general methods is Optimized Perturbation Theory now widely employed in various branches of physics, chemistry, and applied mathematics. The other powerful method is Self-Similar Approximation Theory allowing for quite simple and accurate summation of divergent series. These theories share many common features with the method of renormalization group, which is briefly sketched in order to stress the similarities in their ideas and their mutual interconnection.

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Section: Linear Hamiltoniansmentioning

confidence: 99%

Interplay between Approximation Theory and Renormalization Group

Yukalov

2019

Phys. Part. Nuclei

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“…Besides, because the Riccati equation ( 4) has in the relativistic case the same structure as in the nonrelativistic one [21,22], these coupling constants, V i and W i , appear in common with powers of Plank's constant. Therefore the perturbation series must be in reality not only expansions in powers of a screening parameter but also the semiclassical h-expansions, too.…”

Section: The Methodsmentioning

confidence: 99%

“…Recently, a new technique based on a specific quantization conditions has been proposed to get the perturbation series via the semiclassical h-expansions within the one dimensional Schrödinger equation [21,22].…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Perturbation Theory for Radial Klein–gordon Equation With Screened Coulomb Potentials via ℏ-Expansions

Dobrovolska

Tutik

2004

Int. J. Mod. Phys. A

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Abstract. The explicit semiclassical treatment of logarithmic perturbation theory for the bound-state problem within the framework of the radial Klein-Gordon equation with attractive real-analytic screened Coulomb potentials, contained time-component of a Lorentz four-vector and a Lorentz-scalar term, is developed. Based upon h-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues for the Hulthén potential containing the vector part as well as the scalar component are considered.

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“…according to whether the radial or axial modes are excited. For highly excited modes, one could invoke an optimized expansion in powers of h [36]. The number N 0 is also very sensitive to the trap shape, depending on the aspect ratio (31).…”

Section: Transition Amplitudesmentioning

confidence: 99%

Nonlinear coherent modes of trapped Bose-Einstein condensates

Yukalov

Yukalova²,

Bagnato³

2002

Phys. Rev. A

126

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Nonlinear coherent modes are the collective states of trapped Bose atoms, corresponding to different energy levels. These modes can be created starting from the ground state condensate that can be excited by means of a resonant alternating field. A thorough theory for the resonant excitation of the coherent modes is presented. The necessary and sufficient conditions for the feasibility of this process are found. Temporal behaviour of fractional populations and of relative phases exhibits dynamic critical phenomena on a critical line of the parametric manifold. The origin of these critical phenomena is elucidated by analyzing the structure of the phase space. An atomic cloud, containing the coherent modes, possesses several interesting features, such as interference patterns, interference current, spin squeezing, and massive entanglement. The developed theory suggests a generalization of resonant effects in optics to nonlinear systems of Bose-condensed atoms. 03.75. Fi

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A Recursion Technique for Deriving Renormalized Perturbation Expansions for One-Dimensional Anharmonic Oscillator

Cited by 9 publications

References 32 publications

Interplay between Approximation Theory and Renormalization Group

Interplay between Approximation Theory and Renormalization Group

Logarithmic Perturbation Theory for Radial Klein–gordon Equation With Screened Coulomb Potentials via ℏ-Expansions

Nonlinear coherent modes of trapped Bose-Einstein condensates

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