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Register a new userRadial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. II. Quartic and Sextic Anharmonicity Cases
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In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= frac{1}{g^2},hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - 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 $V(r)= r^2 + g^{2(m-1)}, r^{2m}, m=2,3$, 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 $D=1,2,3ldots $ and $g geq 0$, while the relative deviation of the Approximant from the exact eigenfunction is less than $10^{-6}$ for any $r geq 0$.
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A simple method for the calculation of higher orders of the logarithmic perturbation theory for bound states of the spherical anharmonic oscillator is developed. The structure of the perturbation series for energy eigenvalues of the sextic doubly anharmonic oscillator is investigated. The recursion technique for deriving renormalized perturbation expansions is offered.
It is shown that for one-dimensional anharmonic oscillator with potential $V(x)= a x^2+ldots=frac{1}{g^2},hat{V}(gx)$ (and for perturbed Coulomb problem $V(r)=frac{alpha}{r} + ldots = g,tilde{V}(gr)$) the Perturbation Theory in powers of coupling constant $g$ (weak coupling regime) and semiclassical expansion in powers of $hbar^{1/2}$ for energies coincide. %The same is true for strong coupling regime expansion in inverse fractional powers in $g$ of energy. It is related to the fact that the dynamics developed in two spaces: $x (r)$-space and in $gx (gr)$ space, leads to the same energy spectra. The equations which govern dynamics in these two spaces, the Riccati-Bloch equation and the Generalized Bloch(GB) equation, respectively, are presented. It is shown that perturbation theory for logarithmic derivative of wave function in $gx (gr)$ space leads to true semiclassical expansion in powers of $hbar^{1/2}$ and corresponds to flucton calculus for density matrix in path integral formalism in Euclidean (imaginary) time.
The explicit semiclassical treatment of the logarithmic perturbation theory for the bound-state problem for the spherical anharmonic oscillator is developed. Based upon the $hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the quartic anharmonic oscillator are considered.
The spectral problem for O(D) symmetric polynomial potentials allows for a partial algebraic solution after analytical continuation to negative even dimensions D. This fact is closely related to the disappearance of the factorial growth of large orders of the perturbation theory at negative even D. As a consequence, certain quantities constructed from the perturbative coefficients exhibit fast inverse factorial convergence to the asymptotic values in the limit of large orders. This quantum mechanical construction can be generalized to the case of quantum field theory.
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 $hbar$ 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 $hbar g^2$. 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 $hbar$ for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A 2-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy $sim 10^{-6}$ locally and unprecedented accuracy $sim 10^{-9}-10^{-10}$ in energy for any $g^2 geq 0$. A generalization to the radial quartic oscillator is briefly discussed.
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