Large-Order Strong Coupling Perturbation Coefficients for Anharmonic Oscillators

A large-order formula for the perturbation coefficients of the strong-couplingperturbation expansion for anharmonic oscillators with Hamiltonian H = p ² +x ² + x ^2m is derived and parameters in thisformula are determined for the ground and first excited states of the quartic,sextic, octic, and decadic oscillators (m = 2, 3, 4, 5).     

An asymptotic expansion for energy eigenvalues of anharmonic oscillators

In the present contribution, we derive an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V(x)=κx^2q+ωx²,q=2,3,…$V\left(x\right)=\kappa x^{2q}+\omega x^2,q=2,3,\dots$ as the energy level n$n$ approaches infinity. The asymptotic expansion is obtained using the WKB theory and series reversion. Furthermore, we construct an algorithm for computing the coefficients of the asymptotic expansion for quartic anharmonic oscillators, leading to an efficient and accurate computation of the energy values for n≥6$n\ge 6$.     

On the energy spectra of one-dimensional anharmonic oscillators

In this paper we present explicit and simple analytical formulae for the energy eigenvaluesE _n (λ) of one-dimensional anharmonic oscillators characterized by the potentials 1/2mω ² x ²+λx ^2α withα=2, 3 and 4. A simple intuitive criterion supplemented by the requirement of correct asymptotic behaviour, has been employed in arriving at the formulae. Our energy values over a wide range ofn andλ are in good agreement with the numerical values computed by earlier workers through very elaborate techniques. To our knowledge this is the first time that formulae of such wide validity have been given. The results for pure power oscillators are trivially obtained by going over to theω→0 limit. Approximate analytic expressions for the low order even moments ofx are also given.     

A class of exact solutions for doubly anharmonic oscillators

The solution of a difference equation in the form of an infinite continued fraction is used to obtain a class of exact solutions for the eigenfunctions and eigenvalues of doubly anharmonic oscillators described by potentials of the type (1/2)²x²+(1/4)x⁴+(1/6)x⁶, n>0, provided certain constraints on the couplings are satisfied. The class is denumerably infinite but not complete.     

Matrix moment methods in perturbation theory,boson quantum field models,and anharmonic oscillators

A matrix moment problem is considered in connection with anyx ^2m (m=2, 3, 4, ...) anharmonic oscillator as well as the (:^2m (x):g(x))₂ (m=2, 3) field theory models, whose Rayleigh-Schrödinger perturbation expansions for the ground state eigenvalue are known to diverge. The approximants related to such a problem are proven to converge to the eigenvalue, when applied to an expansion of the Brillouin-Wigner type. These approximants, whose construction involves only matrix elements occurring in the Rayleigh-Schrödinger expansion, are the approximants of aJ-type matrix continued fraction, i.e. the [N–1,N] matrix Padé approximants. The explicit analytical expression of matrix continued fraction is found in the anharmonic oscillators case.     

The energy levels for simplified anharmonic oscillators

The energy levels for quantum mechanical oscillators with interaction imitatingx (for integer >2) are found by perturbative methods in finite number of dimensions. It is argued that in the limit of infinite dimensional space the coefficients in the expansion for the energy of theith level are growing with the perturbation ordern like . For the ground state (i=1) this reproduces estimates established for anharmonic oscillators.     

Strong coupling expansion for classical statistical dynamics

We discuss the simple, randomly driven systemdx/dt = –x –x³ +f(t), wheref(t) is a Gaussian random function or stirring force with f(t)f(t) = (t – t). We show how to obtain approximately the coefficients of the expansion of the equal-time Green's functions as power series in (1/R)ⁿ, whereR is the internal Reynolds number ()^1/2/, by using a new expansion for the path integral representation of the generating functional for the correlation functions. Exploiting the fact that the action for the randomly driven system is related to that of a quantum mechanical anharmonic oscillator with Hamiltonianp ²/2 +m ² x ²/2 +vx ⁴ +x ⁶/2, we evaluate the path integral on a lattice by assuming that thex ⁶ term dominates the action. This gives an expansion of the lattice theory Green's functions as power series in 1/(a)^1/3, wherea is the lattice spacing. Using Padé approximants to extrapolate toa = 0, we obtain the desired large-Reynolds-number expansion of the two-point function.Supported financially by the National Science Foundation and the U.S. Department of Energy.     

On the Gibbs states for one-dimensional lattice Boson systems with a long-range interaction

We consider an infinite chain of interacting quantum (anharmonic) oscillators. The pair potential for the oscillators at lattice distanced is proportional to {d ²[log(d+1)]F(d)}^–1 where _rZ [rF(r)]^–1 < . We prove that for any value of the inverse temperature> 0 there exists a limiting Gibbs state which is translationally invariant and ergodic. Furthermore, it is analytic in a natural sense. This shows the absence of phase transitions in the systems under consideration for any value of the thermodynamic parameters.     

Approximate vacuums in (:φ2m :g(x))2 field theories and perturbation series

A class of perturbation problems is considered, in which the Rayleigh-Schrödinger perturbation series for the ground state eigenvalue and eigenvector are presumed to diverge. This class includes the (:^2m :g(x))₂, (m=2, 3) quantum field theory models and the quantum mechanical anharmonic oscillator. It is shown that, using matrix elements and vectors which occur in the series coefficients, one may construct convergent approximants to the eigenvalue and eigenvector. Results of a calculation of the ground state energy of thex ⁴ anharmonic oscillator are given.Supported in part by the National Research Council of Canada.     

Energy levels of λx 2k anharmonic oscillators using the quantum normal form

The ground state and first few excited energy levels of the generalized anharmonic oscillator defined by the HamiltonianH=–d ²/dx ²+x ²+x ^2k (k=3, 4,...) have been calculated by employing the method of quantum normal form, which is the quantum mechanical analogue of the classical Birkhoff-Gustavson normal form. The present energy eigenvalues are consistent with other tabulations of the energy levels.     

A Phase Transition in a Quantum Crystal with Asymmetric Potentials

A translation invariant system of interacting quantum anharmonic oscillators indexed by the elements of a simple cubic lattice is considered. The anharmonic potential is of general type, which in particular means that it might have no symmetry. For this system, we prove that the global polarization (obtained in the thermodynamic limit) gets discontinuous at a certain value of the external field provided d ≥ 3, and the particle mass as well as the interaction intensity are big enough. The proof is based on the representation of local Gibbs states in terms of path measures and thereby on the use of the infrared estimates and the Garsia–Rodemich–Rumsey inequality.      

Dynamical and thermal aspects of relativistic heavy-ion collisions

Within a covariant BUU-approach we simulate heavyion collisions at various bombarding energies from 400 MeV/u to 1 GeV/u. We evaluate locally the energymomentum tensorT ^v (x), and extract pressures, energydensities and temperatures. The connection of these thermodynamical quantities to experimental observables and their sensitivity to the equation of state is discussed. Furthermore, we investigate the question of local equilibration and evaluate the entropy produced in these reactions.Dedicated to Prof. Dr. P. Kienle on the occasion of his 60th birthday. Work supported by BMFT and GSI Darmstadt     

A new variational perturbation method for double-well oscillators

We propose a variational perturbation method based on the observation that eigenvalues of each parity sector of both the anharmonic and double-well oscillators are approximately equi-distanced. The generalized deformed algebra satisfied by the invariant operators of the systems provides well defined Hilbert spaces to both of the oscillators. There appears a natural expansion parameter defined by the ratio of length scales of the trial wavefunctions. The energies of the ground state and the first order excited state, in the zeroth order variational approximation, are obtained with errors <10⁻²% for vast range of the coupling strength for both oscillators. An iterative formula is presented which perturbatively generates higher order corrections from the lower order invariant operators and the first order correction is explicitly given.     

Asymptotic time behavior of correlation functions. II. Kinetic and potential terms

On the basis of the mode-coupling theory we obtain the long-time behavior t ^–d/2 for the kinetic, potential, and cross-terms in the Green-Kubo integrands, expressed completely in terms of transport coefficients and thermodynamic quantities. All two-mode amplitudes are explicitly evaluated in terms of measurable quantities such as specific heats, thermal expansion coefficients, etc.     

Modified perturbation series for the anharmonic oscillator using linearization technique

The linearization technique of random phase approximation is applied to the anharmonic oscillator to find a modified perturbation series. It is shown that for the anharmonic termλx ⁴, the ground state energyE ₀ upto the second order of perturbation is given byE ₀=(35/48) (3/4)^1/3 λ ^1/3 asλ→∞.     

Step Operators for the Quasi-exactly Solvable Systems

In this paper, we investigate the step operators for the quasi-exactly solvable problems. We also discuss the commutation relations between the step operators and the Hamiltonian of the quasi-exactly solvable system. After obtaining the general results, we take the anharmonic oscillators with x ⁶ anharmonicity in quasi-exactly solvable problems as examples to give the specific forms of step operators.     

Energy eigenvalues for anharmonic and double-well oscillators with even power polynomial potential

Energies for different states of anharmonic and double-well oscillators described by V(x) = Σ_{j = 1}^s(a_2j/2j)x^2j have been determined using the renormalized hypervirial-Padé scheme. A comparison of the results with available exact values shows that the method is quite successful for anharmonic oscillators but fails in the case of double-well oscillators. Also included is the discussion of the dependence of the energy on various parameters.     

非谐振子势的精确解和双波函数描述

求解了非谐振子势V(x)=x²/2+g/2x²的本征方程，给出了精确的能谱方程和归一化波函数.应用双波函数理论，得到了在非谐振子势场中单粒子运动状态的力学量的时间演化方程. 关键词：     

Fluctuations and lack of self-averaging in the kinetics of domain growth

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t) ^d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeL ^d of the system. This lack of self-averaging is tested for both the Ising model and the ⁴ model on the square lattice. Both models exhibit the same lawl(t)=(Rt) ^x withx=1/2, although the ⁴ model has soft walls. However, spurious results withx1/2 are obtained if bad pseudorandom numbers are used, and if the numbern of independent runs is too small (n itself should be of the order of 10³). We also predict a critical singularity of the rateR(1–T/T _c) ^v(z–1/x),v being the correlation length exponent,z the dynamic exponent.Also quenches to the critical temperatureT _c itself are considered, and a related lack of self-averaging in equilibrium computer simulations is pointed out for quantities sampled from thermodynamic fluctuation relations.     

Anharmonic Oscillator and Double-Well Potential: Approximating Eigenfunctions

A simple uniform approximation of the logarithmic derivative of the ground state eigenfunction for both the quantum-mechanical anharmonic oscillator and the double-well potential given by V=m ² x ²+g x ⁴ at arbitrary g ≥ 0 for m ²>0 and m ²<0, respectively, is presented. It is shown that if this approximation is taken as unperturbed problem it leads to an extremely fast convergent perturbation theory Mathematics Subject Classifications(2000) 34L40, 34B08, 41A99