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Register a new userSolving the Anharmonic Oscillator: Tuning the Boundary Condition
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We outline a remarkably efficient method for generating solutions to quantum anharmonic oscillators with an x^{2M} potential. We solve the Schroedinger equation in terms of a free parameter which is then tuned to give the correct boundary condition by generating a power series expansion of the wavefunction in x and applying a modified Borel resummation technique to obtain the large x behaviour. The process allows us to calculate energy eigenvalues to an arbitrary level of accuracy. High degrees of precision are achieved even with modest computing power. Our technique extends to all levels of excitation and produces the correct solution to the double well oscillators even though they are dominated by non-perturbative effects.
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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 $hbar$ 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.
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.
The energy eigenvalues of the anharmonic oscillator characterized by the cubic potential for various eigenstates are determined within the framework of the hypervirial-Pade summation method. For this purpose the E[3,3] and E[3,4] Pade approximants are formed to the energy perturbation series and given the energy eigenvalues up to fourth order in terms of the anharmonicity parameter $lambda$.
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 $r geq 0$.
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.
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