Strong coupling regime of the anharmonic oscillator from perturbation theory and application to the Φ44 model

1978 ◽  
Vol 78 (1) ◽  
pp. 107-109 ◽  
Author(s):  
B. Bonnier
Keyword(s):  
Perturbation Theory ◽  
Strong Coupling ◽  
Anharmonic Oscillator ◽  
Strong Coupling Regime

scholarly journals Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

2020 ◽  
Vol 35 (01) ◽  
pp. 2050005
Author(s):  
J. C. del Valle ◽  
A. V. Turbiner
Keyword(s):  
Perturbation Theory ◽  
Anharmonic Oscillator ◽  
Strong Coupling Regime ◽  
Weak Coupling Regime ◽  
Fractional Powers ◽  
Anharmonic Oscillators ◽  
Relative Deviation ◽  
Variational Calculations ◽  
Compact Approximation ◽  
Semiclassical Expansion

In our previous paper I (del Valle–Turbiner, 2019) a formalism was developed to study the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text]. It was based on the Perturbation Theory (PT) in powers of [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) in both [Formula: see text]-space and in [Formula: see text]-space, respectively. As a result, the Approximant was introduced — a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations, it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper, the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials [Formula: see text], [Formula: see text], respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8–12 figures for any [Formula: see text] and [Formula: see text], while the relative deviation of the Approximant from the exact eigenfunction is less than [Formula: see text] for any [Formula: see text].

φ2k OSCILLATOR IN THE STRONG COUPLING LIMIT

1993 ◽  
Vol 08 (29) ◽  
pp. 5129-5140 ◽  
Author(s):  
L.D. KORSUN ◽  
A.N. SISSAKIAN ◽  
I.L. SOLOVTSOV
Keyword(s):  
Perturbation Theory ◽  
Strong Coupling ◽  
Anharmonic Oscillator ◽  
Strong Coupling Limit ◽  
Functional Integrals ◽  
Nonperturbative Method ◽  
Gaussian Quadratures ◽  
Standard Perturbation Theory

A nonperturbative method for calculating functional integrals is proposed. Only Gaussian quadratures and only those types of diagrams which occur in the standard perturbation theory are used for the formulation of this method. The proposed approach is used for the consideration of the φ2k anharmonic oscillator in the strong coupling limit.

scholarly journals Inconsistencies of the New No-Boundary Proposal

Universe ◽  
2018 ◽  
Vol 4 (10) ◽  
pp. 100 ◽  
Author(s):  
Job Feldbrugge ◽  
Jean-Luc Lehners ◽  
Neil Turok
Keyword(s):  
Boundary Condition ◽  
Perturbation Theory ◽  
Strong Coupling ◽  
Path Integral ◽  
Quantum Cosmology ◽  
Einstein Gravity ◽  
Strong Coupling Regime ◽  
Recent Modification ◽  
Circular Contour ◽  
Made In

In previous works, we have demonstrated that the path integral for real, Lorentzian four-geometries in Einstein gravity yields sensible results in well-understood physical situations, but leads to uncontrolled fluctuations when the “no boundary” condition proposed by Hartle and Hawking is imposed. In order to circumvent our result, new definitions for the gravitational path integral have been sought, involving specific choices for a class of complex four-geometries to be included. In their latest proposal, Diaz Dorronsoro et al. advocate for integrating the lapse over a complex circular contour enclosing the origin. In this note, we show that, like their earlier proposal, this leads to mathematical and physical inconsistencies and thus cannot be regarded as a basis for quantum cosmology. We also comment on Vilenkin and Yamada’s recent modification of the “tunneling" proposal, made in order to avoid the same problems. We show that it leads to the breakdown of perturbation theory in a strong coupling regime.

scholarly journals CONVERGENT SEQUENCES OF PERTURBATIVE APPROXIMATIONS FOR THE ANHARMONIC OSCILLATOR I: HARMONIC APPROACH

1996 ◽  
Vol 11 (31) ◽  
pp. 5587-5606 ◽  
Author(s):  
B. BELLET ◽  
P. GARCIA ◽  
A. NEVEU
Keyword(s):  
Perturbation Theory ◽  
Strong Coupling ◽  
Anharmonic Oscillator ◽  
Convergent Sequence ◽  
Strong Coupling Limit ◽  
Field Theories ◽  
New Techniques ◽  
Numerical Evidence ◽  
Convergent Sequences

We present numerical evidence that a simple variational improvement of the ordinary perturbation theory of the quantum anharmonic oscillator can give a convergent sequence of approximations even in the extreme strong coupling limit, the purely anharmonic case. Some of the new techniques of this paper can be extended to renormalizable field theories.

scholarly journals Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. I. Generalities, cubic anharmonicity case

2019 ◽  
Vol 34 (26) ◽  
pp. 1950143 ◽  
Author(s):  
J. C. del Valle ◽  
A. V. Turbiner
Keyword(s):  
Perturbation Theory ◽  
Anharmonic Oscillator ◽  
Logarithmic Derivative ◽  
Variational Calculus ◽  
Strong Coupling Regime ◽  
Coupling Constant ◽  
Dimension Formula ◽  
Loop Expansion ◽  
Exact Ground State ◽  
Semiclassical Expansion

For the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text] the perturbation theory (PT) in powers of coupling constant [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) is developed constructively in [Formula: see text]-space and in [Formula: see text]-space, respectively. The Riccati–Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate [Formula: see text]-space and in [Formula: see text]-space, respectively, exploring the logarithmic derivative of wave function [Formula: see text]. It is shown that PT in powers of [Formula: see text] developed in RB equation leads to Taylor expansion of [Formula: see text] at small [Formula: see text] while being developed in GB equation leads to a new form of semiclassical expansion at large [Formula: see text]: it coincides with loop expansion in path integral formalism. In complementary way PT for large [Formula: see text] developed in RB equation leads to an expansion of [Formula: see text] at large [Formula: see text] and developed in GB equation leads to an expansion at small [Formula: see text]. Interpolating all four expansions for [Formula: see text] leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at [Formula: see text] for any coupling constant [Formula: see text] and dimension [Formula: see text]. As a concrete application, the low-lying states of the cubic anharmonic oscillator [Formula: see text] are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is [Formula: see text] for [Formula: see text] for coupling constant [Formula: see text] and dimension [Formula: see text] In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7–8 s.d. for [Formula: see text] and [Formula: see text]

Strong Coupling: Polariton Bose Condensation

2017 ◽  
Author(s):  
Alexey V. Kavokin ◽  
Jeremy J. Baumberg ◽  
Guillaume Malpuech ◽  
Fabrice P. Laussy
Keyword(s):  
Strong Coupling ◽  
Room Temperature ◽  
Elevated Temperatures ◽  
Dissipative System ◽  
Bose Condensation ◽  
Einstein Condensation ◽  
Strong Coupling Regime ◽  
Stimulated Scattering ◽  
Exciton Polaritons ◽  
Bose Einstein

In this Chapter we address the physics of Bose-Einstein condensation and its implications to a driven-dissipative system such as the polariton laser. We discuss the dynamics of exciton-polaritons non-resonantly pumped within a microcavity in the strong coupling regime. It is shown how the stimulated scattering of exciton-polaritons leads to formation of bosonic condensates that may be stable at elevated temperatures, including room temperature.

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