One-dimensional hypervirial theorems in density functional theory

For a restricted class of operators hypervirial theorems are established involving wavefunctions ψ_i for N fermions which move independently in a common external one-dimensional potential V(x). Using this class it is possible to perform the summation over states ψ_i yielding exact relations between ground state density ? and kinetic energy density ?_k and potential V(x). It is checked to what extent the Thomas-Fermi expressions for ? and ?_k satisfy these relations.     

Quantum hypervirial theorems

The quantum mechanical hypervirial theorems (HVT) are given by the expectation values of a commutator between the virial operator W and the system Hamiltonian H. We propose application of the HVT to testing and improving approximate solutions of the Schroedinger equations. This is especially relevant for scattering states, where simple testing criteria are not readily available. The HVT, with judicious choices for W, can provide a criterion to test the accuracy of approximate solutions, both for the bound excited states and scattering states, and also ways to determine an optimal set of parameter values as the wave function improves.     

Perturbation theory of odd anharmonic oscillators

We study the perturbation theory forH=p ²+x ²+x ²ⁿ⁺¹,n=1, 2, .... It is proved that when Im0,H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schrödinger perturbation expansion, and admits an analytic continuation to Im=0 where it can be interpreted as a resonance of the problem.Partially supported by G.N.F.M., C.N.R.     

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.     

Partial hypervirial theorems for atomic-molecular systems

A family of partial hypervirial theorems for physical properties of pairs of particles in three-and four-particle atomic-molecular systems is considered. The partial hypervirial theorems generalize the partial virial theorems proposed earlier. The sum rule is formulated, according to which the expectation values of the derivatives with respect to the interparticle distances are related to the products of the charges of pairs of the particles. This rule is used to check the accuracy of the calculation of variational wave functions for the positronium ion e ^? e ^? e ⁺. It is shown that the sum rule is two orders of magnitude more sensitive to the inaccuracy of the calculation of the wave function than the partial virial theorems and is five or six orders of magnitude more sensitive to it than the conventional virial theorem.     

On the anharmonic oscillator in quantum mechanics and in field theory

A modified Rayleigh-Schrödinger perturbation method is used to derive explicit expansions for the eigenvalues and eigensolutions of the anharmonic oscillator. We then point out the dual relationship between the anharmonic oscillator and the Schrödinger equation for a Yukawa potential. Finally we consider an application of the method to a field-theoretic Hamiltonian, since the anharmonic oscillator plays a dominant role in many field-theoretic models.     

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.     

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.     

1/N-expansion for the anharmonic oscillator

We investigate the properties of the 1/N-expansion for the quartic anharmonic oscillator in quantum mechanics. The first seven terms of the expansion for the energy ground and first excited levels are obtained analytically. We have found also the large-order behaviour of the 1/N-expansion coefficients in closed form and convinced ourselves that the asymptotic series obtained is Borel summable. We use the formulae derived to find the first seven coefficients of the perturbative expansion in powers of the coupling constant in the case of the double-well potential for arbitrary number of componentsN. These exact expressions enable us to guess the large-order behaviour of the perturbative coefficients forN=0,1, ..., 4. At the end we give an example of summing the asymptotic series in powers of 1/N applying the Padé-Borel method.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.We thank Drs. D. I. Kazakov, V. K. Mitrjushkin and O. V. Tarasov for their interest and for useful discussions. One of us (M.A.S.) is particularly grateful to Dr. Jií Niederle for his kind hospitality during the Symposium at Bechyn.     

Pade approximants and the anharmonic oscillator

The diagonal Padé approximants of the perturbation series for the eigenvalues of the anharmonic oscillator (a βκ¹ perturbation of p² + κ²) converge to the eigenvalues.     

Exact energy eigenvalues for the doubly anharmonic oscillator

The exact energy eigenvalues for the potentialV(r) = 1/2 ² r ² + 1/4r⁴ +1/6 r⁶ and the conditions for their occurrence have been determined.     

A supersymmetric anharmonic oscillator

We consider the supersymmetric model obtained from the Wess-Zumino model with interaction, with the help of a dimensional reduction. The solution is found of the corresponding classical problem with a precision up to terms of the third order of smallness. A quantum model of such a system is constructed. Its spectrum is found in the first and second order of perturbation theory. The supersymmetric finite-dimensional model under consideration proves to be extremely close to the usual anharmonic oscillator.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 28–32, May, 1990.     

非谐振子Gazeau-Klauder与Klauder-Perelomov相干态

利用Kinani-Daoud方法构造了非谐振子势的Gazeau-Klauder(GK)相干态和Klauder-Perelomov(KP)相干态，表明两种相干态在非线性谐振子势下具有完全不同的形式，并对两种相干态的完备性以及各自构成的Hilbert空间进行了讨论. 对相干态的Mandel Q参数的研究表明：GK相干态服从亚Poisson统计分布，KP相干态服从超Poisson统计分布. 关键词： 非谐振子 Gazeau-Klauder相干态 Klauder-Perelomov相干态     

A new perturbative approach for the anharmonic oscillator

A new perturbative approach is presented and applied to the anharmonic oscillator. Estimating the large-order behaviour by means of the Lipatov method, the new perturbation expansion turns out to be convergent for the partition function at non-zero temperature. Although the new perturbation expansion is not convergent for the ground-state energy, it nevertheless provides accurate approximations, even in situations which cannot be treated by standard perturbation techniques, like the strong coupling regime.     

Residue-squaring diagonalisation method and the anharmonic oscillator

A recently-formulated residue-squaring method for perturbation problems is subjected to an exacting test in its application to the problem of diagonalising the Hamiltonian of the nonlinear oscillator with quartic anharmonicity. Unlike other methods, this new iterative diagonalisation method enables several eigenvalues to be calculated simultaneously with little more labour than for a single eigenvalue. Values obtained for the four lowest even-parity levels of the anharmonic oscillator from just two or three iterations are shown to agree well with earlier accurate calculations. An approximate analytical formula for the energy levels is also presented.     

Functional renormalization group for quantized anharmonic oscillator

Functional renormalization group methods formulated in the real-time formalism are applied to the O(N) symmetric quantum anharmonic oscillator, considered as a 0 + 1 dimensional quantum field-theoric model, in the next-to-leading order of the gradient expansion of the one- and two-particle irreducible effective action. The infrared scaling laws and the sensitivity-matrix analysis show the existence of only a single, symmetric phase. The Taylor expansion for the local potential converges fast while it is found not to work for the field-dependent wavefunction renormalization, in particular for the double-well bare potential. Results for the gap energy for the bare anharmonic oscillator potential hint on improving scheme-independence in the next-to-leading order of the gradient expansion, although the truncated perturbation expansion in the bare quartic coupling provides strongly scheme-dependent results for the infrared limits of the running couplings.     

Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory

We introduce a generalization of Wick-ordering which maps the anharmonic oscillator (AO) Hamiltonian for mass m and coupling λ exactly into a “Wick-ordered” Hamiltonian with an effective mass M which is a simple analytic function of λ and m. The effective coupling $\text{Λ =}\text{λ}\text{M}{}^3$ is bounded. We transform the AO perturbation series in λ into one in Λ. This series may then be summed using Borel summation methods. We also introduce a new summation method for the AO series (which is a practical necessity to obtain accurate energy levels of the excited states). We obtain a numerical accuracy for $\text{(E}{}_{\mathrm{PT}}\text{? E}{}_{\mathrm{exact}}\text{)}\text{E}{}_{\mathrm{exact}}$ of at least 10^?7 (using 20 orders of perturbation theory) and 10^?3 (using only 2 orders of perturbation theory) for all couplings and all energy levels of the anharmonic oscillator. The methods are applicable also to the double-well potential (DWP, the AO with a negative mass-squared). The only change is that now the effective coupling is unbounded as λ → 0. The series in Λ is, however, still summable. The relative accuracy in the energy levels for 20 orders of perturbation theory varies from 10^?7 for large coupling to 1% at λ = 0.1 and to 10% at λ = .05. We also present results for the sextic oscillator.