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

Valle, Juan Carlos del; Turbiner, A. V.

doi:10.1142/s0217751x20500050

Cited by 8 publications

(4 citation statements)

References 21 publications

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“…It manifests a solution of the problem of ground state of cubic radial anharmonic oscillator as well as its low-lying excited states! Based on this formalism similar accuracy is reached for quartic and sextic radial anharmonic oscillators that will be presented in subsequent paper [44].…”

supporting

confidence: 56%

Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. I. Generalities, Cubic Anharmonicity Case

del Valle,

Turbiner

2019

Preprint

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For the general D-dimensional radial anharmonic oscillator with potential V (r) = 1 g 2 V (gr) the Perturbation Theory (PT) in powers of coupling constant g (weak coupling regime) and in inverse, fractional powers of g (strong coupling regime) is developed constructively in r-space and in (gr) space, respectively. The Riccati-Bloch (RB) equation and Generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate r-space and in (gr)-space, respectively, exploring the logarithmic derivative of wavefunction y. It is shown that PT in powers of g developed in RB equation leads to Taylor expansion of y at small r while being developed in GB equation leads to a new form of semiclassical expansion at large (gr): it coincides with loop expansion in path integral formalism. In complementary way PT for large g developed in RB equation leads to an expansion of y at large r and developed in GB equation leads to an expansion at small (gr). Interpolating all four expansions for y leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at r ∈ [0, ∞) for any coupling constant g ≥ 0 and dimension D > 0. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. As a concrete application the low-lying states of the cubic anharmonic oscillator V = r 2 + gr 3 are considered. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is 10 −4 for r ∈ [0, ∞) for coupling constant g ≥ 0 and dimension D = 1, 2, . . .. In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7-8 s.d. for g ≥ 0 and D = 1, 2, . . ..

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supporting

confidence: 56%

Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. I. Generalities, Cubic Anharmonicity Case

del Valle,

Turbiner

2019

Preprint

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“…Since the GB equation ( 14) and the true semiclassical expansion (15) can be written for any anharmonic oscillator bounded from below, the approximate eigenfunctions of type (18) can be built. Evidently, similar procedure can be realized for any radial anharmonic oscillator, see [10].…”

Section: Discussionmentioning

confidence: 84%

“…The results for the ground state for different D and g 2 are presented in table 1. All digits for energies in table 1 are exact: it was checked in Lagrange mesh calculations, also by calculating the first correction to the variational energy E 2 and making comparison with variational energies obtained with more parametric generalization of (22), see [10], part II. Details will be presented elsewhere as well as a generalization for the excited states.…”

Section: Radial Quartic Anharmonic Oscillatormentioning

confidence: 99%

Anharmonic oscillator: a solution

Turbiner¹,

Valle²

2021

J. Phys. A: Math. Theor.

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It is shown that for the one-dimensional quantum anharmonic oscillator with potential V(x) = x 2 + g 2 x 4 the perturbation theory (PT) in powers of g 2 (weak coupling regime) and the semiclassical expansion in powers of ℏ for energies coincide. It is related to the fact that the dynamics in x-space and in (gx)-space corresponds to the same energy spectrum with effective coupling constant ℏg 2. Two equations, which govern the dynamics in those two spaces, the Riccati–Bloch (RB) and the generalized Bloch (GB) equations, respectively, are derived. The PT in g 2 for the logarithmic derivative of wave function leads to PT (with polynomial in x coefficients) for the RB equation and to the true semiclassical expansion in powers of ℏ for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A two-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in x-space with unprecedented accuracy ∼10−6 locally and unprecedented accuracy ∼10−9–10−10 in energy for any g 2 ⩾ 0. A generalization to the radial quartic oscillator is briefly discussed.

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“…Generally, the anharmonic potential can be expressed in terms of even-power polynomials. Nevertheless, in our analysis, the anharmonic potential is different from the previous works, in which authors mainly focus on wave functions and eigenvalues of quartic, sextic and decatic even-power polynomial anharmonic potential [11,12].…”

Section: Anharmonic Potentialmentioning

confidence: 96%

Quantum infinite square well as a limiting case of higher order anharmonic oscillator

Salamu

2023

Phys. Scr.

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

Cited by 8 publications

References 21 publications

Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. I. Generalities, Cubic Anharmonicity Case

Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. I. Generalities, Cubic Anharmonicity Case

Anharmonic oscillator: a solution

Quantum infinite square well as a limiting case of higher order anharmonic oscillator

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