Asymptotic behavior of the eigenenergies of anharmonic oscillators V(x) = x2N + bx2

2002 ◽  
Vol 80 (9) ◽  
pp. 959-968 ◽  
Author(s):  
A Nanayakkara ◽  
V Bandara
Keyword(s):  
Asymptotic Behavior ◽  
Energy Level ◽  
Asymptotic Expansions ◽  
Coupling Strength ◽  
Anharmonic Oscillator ◽  
Level Spacing ◽  
Anharmonic Oscillators ◽  
Analytic Expressions ◽  
Asymptotic Energy

Analytic semiclassical energy expansions of the anharmonic oscillator V(x) = x2N + bx2 are obtained for arbitrary N. These expressions contain the parameters b and N of the potential explicitly. Analytic expressions for energy level spacing are obtained and used to study the behavior of the eigenenergy level spacing for large energies. These expressions show that asymptotic energy level spacing of the potential V(x) = x2N + bx2 increases with the coupling strength b for N = 2 and 3, whereas it decreases for N > 3. Validity of the asymptotic expansions for noninteger N is discussed. PACS Nos.: 03.65Ge, 03.65Sq, 02.30Mv

Asymptotic behavior of eigen energies of non-Hermitian cubic polynomial systems

2007 ◽  
Vol 85 (12) ◽  
pp. 1473-1480 ◽  
Author(s):  
A Nanayakkara
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial System ◽  
Expansion Method ◽  
Polynomial Systems ◽  
Level Spacing ◽  
Spacing Distribution ◽  
Cubic Polynomial ◽  
Asymptotic Energy ◽  
04.20 Jb ◽  
Energy Expansion

The asymptotic behavior of the eigenvalues of a non-Hermitian cubic polynomial system H = (P2/2) + µx3 + ax2 + bx, where µ, a, and b are constant parameters that can be either real or complex, is studied by extending the asymptotic energy expansion method, which has been developed for even degree polynomial systems. Both the complex and the real eigenvalues of the above system are obtained using the asymptotic energy expansion. Quantum eigen energies obtained by the above method are found to be in excellent agreement with the exact eigenvalues. Using the asymptotic energy expansion, analytic expressions for both level spacing distribution and the density of states are derived for the above cubic system. When µ = i, a is real, and b is pure imaginary, it was found that asymptotic energy level spacing increases with the coupling strength a for positive a while it decreases for negative a. PACS Nos.: 03.65.Ge, 04.20.Jb, 03.65.Sq, 02.30.Mv, 05.45

Asymptotic behavior of eigenenergies of nonpolynomial oscillator potentials V(x) = x2N + (λx m1)/(1 + gx m2)

2012 ◽  
Vol 90 (6) ◽  
pp. 585-592 ◽  
Author(s):  
Asiri Nanayakkara
Keyword(s):  
Asymptotic Behavior ◽  
Power Series ◽  
Recurrence Relations ◽  
Energy Spectra ◽  
Action Variable ◽  
Analytic Expressions ◽  
Quantum Action ◽  
Asymptotic Energy ◽  
Polynomial Potentials ◽  
Positive Integers

Analytic semiclassical energy expansions of nonpolynomial oscillator (NPO) potentials V(x) = x2N + (λx[Formula: see text])/(1 + gx[Formula: see text]) are obtained for arbitrary positive integers N, m1, and m2, and the real parameters λ and g using the asymptotic energy expansion (AEE) method. Because the AEE method has been previously developed only for polynomial potentials, the method is extended with new types of recurrence relations. It is then applied to the preceding general NPO to obtain expressions for quantum action variable J in terms of E and the parameters of the potential. These expansions are power series in energy and the coefficients of the series contain parameters λ and g explicitly. To avoid the singularities in the potential we only consider the cases where both λ and g are non-negative at the same time. Using the AEE expressions, it is shown that, for certain classes of NPOs, if potentials have the same N, and the same m1 – m2 or m1 – 2m2 then they have the same asymptotic eigenspectra. It was also shown that for certain cases, both λ and –λ as well as g and –g will produce the same asymptotic energy spectra. Analytic expressions are also derived for asymptotic level spacings of general NPOs in terms of λ and g.

ENERGY SPECTRAL STATISTICS IN THE QUANTUM SYSTEM WITH TWO PARTICLES

2004 ◽  
Vol 18 (17n19) ◽  
pp. 2740-2744 ◽  
Author(s):  
SHIPING YANG ◽  
GUOYONG YUAN ◽  
ZHE LI ◽  
HONG CHANG ◽  
DE LIU
Keyword(s):  
Energy Level ◽  
Quantum System ◽  
Quantum Chaos ◽  
Level Spacing ◽  
Tunnelling Effect ◽  
Spectral Statistics

In this paper, the quantum system with two particles is analyzed and the energy level spacing statistics distribution and Δ3-statistic are given. The results show that hard quantum chaos appear in the system with a certain potential. Tunnelling effect develops quantum chaos.

The dependence of the harmonic oscillator energy level spacing on the mass number of nuclei

1983 ◽  
Vol 121 (2-3) ◽  
pp. 91-95 ◽  
Author(s):  
C.B. Daskaloyannis ◽  
M.E. Grypeos ◽  
C.G. Koutroulos ◽  
S.E. Massen ◽  
D.S. Saloupis
Keyword(s):  
Energy Level ◽  
Harmonic Oscillator ◽  
Mass Number ◽  
Level Spacing

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].

THE ASYMPTOTIC ITERATION METHOD FOR THE EIGENENERGIES OF THE ASYMMETRICAL QUANTUM ANHARMONIC OSCILLATOR POTENTIALS $V(x)=\sum_{j=2}^{2\alpha}A_{j}x^{j}$

2007 ◽  
Vol 22 (01) ◽  
pp. 203-212 ◽  
Author(s):  
T. BARAKAT ◽  
O. M. AL-DOSSARY
Keyword(s):  
Rate Of Convergence ◽  
Iteration Method ◽  
Anharmonic Oscillator ◽  
Full Range ◽  
Adjustable Parameter ◽  
Numerical Results ◽  
Asymptotic Iteration Method ◽  
Anharmonic Oscillators ◽  
Parameter Values

The asymptotic iteration method is used to calculate the eigenenergies for the asymmetrical quantum anharmonic oscillator potentials [Formula: see text], with (α = 2) for quartic, and (α = 3) for sextic asymmetrical quantum anharmonic oscillators. An adjustable parameter β is introduced in the method to improve its rate of convergence. Comparing the present results with the exact numerical values, and with the numerical results of the earlier works, it is found that asymptotically, this method gives accurate results over the full range of parameter values Aj.

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