Ядерна фізика та енергетика Nuclear Physics and Atomic Energy   ISSN: 1818-331X (Print), 2074-0565 (Online)   Publisher: Institute for Nuclear Research of the National Academy of Sciences of Ukraine   Languages: Ukrainian, English   Periodicity: 4 times per year   Open access peer reviewed journal

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V. A. Babenko^*

Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

^*Corresponding author. E-mail address: pet2@ukr.net

Abstract: For the quantum quartic anharmonic oscillator with Hamiltonian H = ½(p² + ω²x²) + λx⁴, our study has been further developed using a convergent expansion of the system's wave function in a modified oscillator basis with an adjustable frequency ω₀, i.e., in the complete set of eigenfunctions {φ_n(ω₀;x)} of a reference harmonic Hamiltonian H⁽⁰⁾(ω₀) = ½(p²+ω₀²x²). Our proposed approach enables a significant improvement in the convergence rate of expansions through the additional adaptation of the basis functions to the specific structure of the system being analyzed. Primary attention is given to the rigorous mathematical implementation of the variational scheme, treating the basis frequency ω₀ as a true nonlinear variational parameter subject to optimization. Within the framework of the generalized nonlinear Rayleigh - Ritz variational method, the system's energy is minimized with respect to the nonlinear parameter ω₀ to accurately determine the optimal value of ω₀ for different values of the oscillator coupling constant λ and the variational basis size N. This study provides, for the first time, a fully rigorous numerical implementation of this procedure for the quartic anharmonic oscillator, ensuring precise control and enabling a transition from heuristic parameter selection to a systematic and reproducible algorithmic approach. This rigorous variational justification enables the exploration of the energy functional's structure and the behavior of the nonlinear parameter. It is shown that the proposed method provides standard 10^-8 accuracy in energy calculations, employing a remarkably small basis size of N ≃ 6 - 7, while maintaining high efficiency throughout the strong-coupling region - for all values of λ. The behavior of the optimized frequency parameter ω₀ with respect to the basis size N was also investigated, along with the identification and interpretation of the important effect of the variational plateau - a broad region of variation in the parameter ω₀ where the computed energy remains nearly constant. The system's wave functions were also calculated using the proposed method for a range of λ. The obtained results demonstrate that nonlinear optimization of the basis is a highly effective means of substantially increasing the efficiency of variational calculations, both for energies and for the wave functions.

Keywords: anharmonic oscillator, harmonic oscillator basis, quartic anharmonic oscillator, nonlinear variational method, Rayleigh - Ritz method, quantum field theory.

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