On perturbation theory for an asymmetric anharmonic oscillator

A modified perturbation theory is formulated for an asymmetric anharmonic oscillator based on a choice of the main approximation constructed by analogy with the self-consistent field model. This perturbation theory has a much wider range of application in comparison with the standard approach and considers a finite number of energy levels in the potential well already in the main approximation. This approach is used for the construction of a two-atomic model of a quantum crystal. A quantum analog of the Lindeman criterion is obtained. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 30–40, January, 2009.     

Variational perturbation theory for Fokker-Planck equation with nonlinear drift

We develop a recursive method for perturbative solutions of the Fokker-Planck equation with nonlinear drift. The series expansion of the time-dependent probability density in terms of powers of the coupling constant is obtained by solving a set of first-order linear ordinary differential equations. Resumming the series in the spirit of variational perturbation theory we are able to determine the probability density for all values of the coupling constant. Comparison with numerical results shows exponential convergence with increasing order.     

Renormalized perturbation theory of the Van der Pol oscillator

The renormalized perturbation expansion for the linewidth of the Van der Pol oscillator is extended to fifth order. The results of the previously obtained fourth order are thereby improved for pump strengthsa5. No indication of a breakdown of the expansion, even for strong pumping, is found.     

Variational spiked oscillator

A variational analysis of the spiked harmonic oscillator Hamiltonian –d ²/dr² +r ² +/r ^5/2,>0, is reported. A trial function automatically satisfying both the Dirichlet boundary condition at the origin and the boundary condition at infinity is introduced. The results are excellent for a very large range of values of the coupling parameter, suggesting that the present variational function is appropriate for the treatment of the spiked oscillator in all its regimes (strong, moderate, and weak interactions).     

Anharmonic oscillators energies via artificial perturbation method

A new pseudoperturbative (artificial in nature) methodical proposal [#!ref15!#] is used to solve for Schr?dinger equation with a class of phenomenologically useful and methodically challenging anharmonic oscillator potentials . The effect of the [#!ref4!#,#!ref5!#] Padé approximant on the leading eigenenergy term is studied. Comparison with results from numerical (exact) and several eligible (approximation) methods is made. Received 2 July 1999 and Received in final form 18 November 1999     

A perturbation theory for identifying quasi-coupled states of an anharmonic oscillator

A perturbation theory designed for calculating quasi-coupled or coupled state energies and which is valid at any value of the coupling parameter is constructed in an example of an anharmonic oscillator with quartic nonlinearity.     

Anharmonic oscillator model for first order structural phase transition

Exact solutions for the motion of a classical anharmonic oscillator in the potentialV(φ)=Bφ ² ? |A|φ ⁴ +Cφ ⁶ are obtained in (1 + 1) dimensions. Instanton-like solutions in (imaginary time) which takes the particle from one maximum of the potential to the other are obtained in addition to the usual oscillatory solutions. The energy dependence of the frequencies of oscillation is discussed in detail. This can be used as a model for the first order structural phase transition in the mean field approximation. The high and low temperature behaviour of the static susceptibility is obtained. Finally, a qualitative explanation is offered for the observed central peak in ferroelectrics like SrTiO₂.     

Convergent perturbation expansion for the anharmonic oscillator

We study the ground state as well as the first three excited states of the anharmonic oscillator with anharmonicity λx⁴ for a range of λ = (0, 10) with the first-order logarithmic perturbation iteration method (FOLPIM). This leads to convergent results. The initial choice of the wave function seems only to affect the rate of convergence in the case of the ground state but may critically affect the convergence for the excited states. For large values of λ, convergence is best obtained by choosing the asymptotic solution as the initial “unperturbed” wave function.     

Self-consistent perturbation theory

The density matrix form of Hartree-Fock perturbation theory is developed for the case in which the basis functions themselves are perturbation-dependent. Energy expressions are derived, through second order, for both single and double perturbations.

The theory is applied in the calculation of electric dipole polarizabilities and hyperpolarizabilities for atoms (He, Be) and molecules (H₂, LiH), with excellent results. The high computational efficiency of the method is discussed and possibilities of further development are outlined.     

Quadrupole perturbation of the harmonic oscillator potential

TheSU(3) symmetry of the harmonic oscillator potential is used to calculate exactly the second order energy correction, caused by quadrupole deformations. This can be done in a purely algebraic manner, using the Dalgarno-Schwartz formulation of perturbation theory. Some connections of this method with group theory are discussed and an extension to more general situations is proposed.     

Wronskian perturbation theory

We develop a perturbation method that generalizes an approach proposed recently to treat velocity-dependent quantum-mechanical models. In order to test the present approach we apply it to some simple trivial and nontrivial examples.     

Molecular cluster perturbation theory. I. Formalism

We present second-order molecular cluster perturbation theory (MCPT(2)), a linear scaling methodology to calculate arbitrarily large systems with explicit calculation of individual wave functions in a coupled-cluster framework. This new MCPT(2) framework uses coupled-cluster perturbation theory and an expansion in terms of molecular dimer interactions to obtain molecular wave functions that are infinite order in both the electronic fluctuation operator and all possible dimer (and products of dimers) interactions. The MCPT(2) framework has been implemented in the new SIA/Aces4 parallel architecture, making use of the advanced dynamic memory control and fine-grained parallelism to perform very large explicit molecular cluster calculations. To illustrate the power of this method, we have computed energy shifts, lattice site dipole moments, and harmonic vibrational frequencies via explicit calculation of the bulk system for the polar and non-polar polymorphs of solid hydrogen fluoride. The explicit lattice size (without using any periodic boundary conditions) was expanded up to 1000 HF molecules, with 32,000 basis functions and 10,000 electrons. Our obtained HF lattice site dipole moments and harmonic vibrational frequencies agree well with the existing literature.     

Action-variable perturbation theory

A quantum mechanical perturbation theory based upon the recently introduced quantum action variable is developed and illustrated. Unlike Rayleigh-Schrödinger or other asymptotic theories, this theory can provide a natural solution for the bound states of any potential.