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

@article{Valle2019RadialAO,
title={Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. I. Generalities, cubic anharmonicity case},
author={Juan Carlos del Valle and Alexander V. Turbiner},
journal={International Journal of Modern Physics A},
year={2019}
}
• Published 10 August 2019
• Mathematics
• International Journal of Modern Physics A
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…
5 Citations

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## References

SHOWING 1-10 OF 45 REFERENCES

• Physics
• 1988
Two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential V(x)= 1/2 x2+(g/4)x4 are proposed. The first one is simply a straightforward
Abstract The Rayleigh–Schrodinger perturbation series for the energy eigenvalue of an anharmonic oscillator defined by the Hamiltonian Ĥ ( m ) ( β )= p 2 + x 2 + βx 2 m with m =2, 3, 4, … diverges
• Physics, Mathematics
• 1972
A quantum anharmonic oscillator with a polynomial self‐interaction is defined in coordinate space by a Hamiltonian of the form H = −d2/dx2 + ¼x2 + g[(½x2)N + a(½x2)N−1 + b(½x2)N−2 + ⋯]. Using WKB
A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire
• Physics
• 2016
We develop a new semiclassical approach, which starts with the density matrix given by the Euclidean time path integral with fixed coinciding endpoints, and proceed by identifying classical (minimal
• Physics
• 2017
This is the second paper on the semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding end points. The classical path, interpolating
• Physics
• 1973
This paper is concerned with the nature of perturbation theory in very high order. Specifically, we study the Rayleigh-Schrodinger expansion of the energy eigenvalues of the anharmonic oscillator. We
• Physics
Journal of Mathematical Physics
• 2018
Using three different approaches Perturbation Theory (PT), the Lagrange Mesh Method (LMM), and the variational method, we study the low-lying states of the Yukawa potential. First orders of PT, in