Quartic Perturbation Research Articles

Homoclinic billiard orbitsinside symmetrically perturbed ellipsoids

The billiard motion inside an ellipsoid ofℝ3 is completely integrable. Ifthe ellipsoid is not of revolution, thereare many orbits bi-asymptotic to its majoraxis. The set of bi-asymptotic orbits isdescribed from a geometrical, dynamicaland topological point of view. It containseight surfaces, called separatrices.The splitting of the separatrices under symmetric perturbations of theellipsoid is studied using a symplectic discrete version of thePoincaré-Melnikov method, with special emphasis on the followingsituations:close to the flat limit(when the minor axis of the ellipsoid is small enough), close to the oblatelimit(when the ellipsoid is close to an ellipsoid of revolution around its minoraxis)and close to the prolate limit(when the ellipsoid is close to an ellipsoid of revolution around its majoraxis).It is proved that any non-quadratic entire symmetric perturbation breaksthe integrability and splits the separatrices, although (at least) 16symmetric homoclinic orbits persist. Close to the flat limit, these orbitsbecome transverse under very general polynomial perturbations of theellipsoid.Finally, a particular quartic symmetric perturbation is analysed in greatdetail. Close to the flat and to the oblate limits, the 16 symmetrichomoclinic orbits are the unique primary homoclinic orbits. Close to theprolate limit, the number of primary homoclinic orbits undergoesinfinitely many bifurcations. The first bifurcation curves are computednumerically.The planar and high-dimensional cases are also discussed.

Pseudo-oscillator with a quartic perturbation

PII: S0093-6413(01)00153-7 PSEUDO-OSCILLATOR WITH A QUARTIC PERTURBATION Stawomir Breiter 1 and Antonio Elipe 2. 1Astronomical Observatory of A. Mickiewicz University, 60-286 Poznafi, Poland. 2Grupo de Mec£nica Espacial. Universidad de Zaragoza, 50009 Zaragoza. Spain (Recetved 13 November 2000, accepted for prmt 23 Janua~ 2001) Introduction The problem to be studied in the paper is a two degrees of freedom Hamiltonian with quadratic part perturbed by a quartic potential = ~2(x, y, X, Y) + ~4(x, y), (1) where x, y are coordinates and X, Y their conjugate momenta. More precisely, we discuss the case with

A non-perturbative cumulant expansion method for the grand partition function of quantum systems

We present here a non-perturbative cumulant expansion method for computing the grand partition function of quantum systems. It embeds the classical component of grand partition function exactly and treats the quantum contributions by a systematic cluster expansion of the Boltzmann operator. The cluster expansion Ansatz exploits our recently developed notion of thermal normal ordering and a Wick-like expansion formula, which makes evaluation of the thermal trace particularly easy. The thermal normal ordering also confers a finite expansion structure to the equations for the cluster amplitudes. Our formulation provides manifestly extensive free energy, and works very well over a wide range of temperatures and coupling strengths. As an illustrative application, we have computed the grand partition function of an anharmonic oscillator with a pure quartic perturbation.

Three-Dimensional Quartic Oscillators for the Study of Mode Fluctuation Distribution

The mode fluctuation distribution (MFD) of three-dimensional quartic oscillators coupled by quartic perturbations is numerically studied. The system is continuously transferred from the integrable to the chaotic regime by changing a single coupling strength. The MFD of the three-dimensional systems is found to be a sensitive measure of integrability as in two-dimensional systems. If the system is exactly integrable, the MFD has the characteristic skew distribution. On the other hand, even in the region that the chaotic and integrable volumes are comparable in the phase space, the MFD already becomes almost the Gaussian distribution as in the completely chaotic regime. We also find that the Brody parameter increases gradually without a plateau by increasing the coupling constant. It shows a clear contrast to the two-dimensional system.

Algebraic linearization of dynamics of Calogero type for any Coxeter group

Calogero–Moser systems can be generalized for any root system (including the noncrystallographic cases). The algebraic linearization of the generalized Calogero–Moser systems and of their quadratic (respectively quartic) perturbations are discussed.

Singular reduction of axially symmetric perturbations ofthe isotropic harmonic oscillator

The normal form of an axially symmetric perturbation of the isotropic harmonic oscillator is invariant under a 2-torus action and thus integrable in three degrees of freedom. The reduction of this symmetry is performed in detail, showing how the singularities of the reduced phase space determine the distribution of periodic orbits and invariant 2-tori in the original perturbation. To illustrate these results a particular quartic perturbation is analysed. AMS classification scheme numbers: 34C20, 58F05, 58F30, 70H33, 70J05, 85A05

Mode fluctuation distribution of coupled quartic oscillators

The mode fluctuation distribution (MFD) of a system of two quartic oscillators coupled by a quartic perturbation is numerically studied. The coupling strength serves as a control parameter to simulate the transition from integrable to chaotic regimes. It is demonstrated that even in this potential system the MFD turns out to be a very sensitive measure, as it manifests the same transformation seen in billiard systems. It is characterized by Gaussian and skewed distributions in the regions, known to be chaotic and integrable, respectively, from classical Poincar\'e sections and quantum-mechanical level spacing statistics. In the intermediate regions where the Kol'mogorov-Arnol'd-Moser tori survive, the MFD has various distorted forms.

Some inner product techniques for computing eigenvalues for three-dimensional anharmonic oscillators with quartic and sextic perturbations

An algorithm previously used by the author for calculating the energy eigenvalues for a three-dimensional oscillator with limited range of perturbation parameters, eigenstates and accuracy is revisited and implemented in a more effective way to give excellent accuracy even for higher values of λ and ψ n x , n y , n z ( x, y, z). Fifty coefficients of renormalized perturbation series are calculated, in contrast to the previous works which were limited to 20 coefficients of the perturbation series due to computational facilities. Some of the results calculated by the inner product technique are compared with results produced by other techniques.

Magnetoplasma excitations in anharmonic electron dots.

We have used a classical hydrodynamic approach to study the magnetoplasma excitations in electron dots. The electrons are assumed to be confined within a plane by an anharmonic potential having circular symmetry. As a necessary first step, we develop a general procedure for determining the equilibrium electron distribution within the dot. This information is then used in the hydrodynamic analysis to yield a generalized eigenvalue problem for the magnetoplasma frequencies. Detailed applications are made to dots defined by a parabolic confining potential with a quartic radial perturbation. Contact with experiment is achieved by means of the calculated power absorption, which is found to be in good agreement with the observed far-infrared transmission spectra. In particular, the hydrodynamic model provides an explanation of the weak satellite, observed to track the high-frequency center-of-mass mode, in terms of the coupling of parabolic modes induced by the potential anharmonicity.

Quantum theory of anharmonic oscillator: Energy levels of a three-dimensional oscillator with quartic anisotropic perturbation

The energy levels of the Schrödinger equation for a three-dimensional oscillator with quartic perturbation, V(x,y,z)= 1/2[x2+y2+z2]+λ[axxx4+ayyy4 +azz z4 +2axyx2y2+2axzx2z2+2ayz y2z2], are calculated using matrix diagonalization for various values of the perturbation parameters and state numbers (nx,ny,nz).

Systematic nonperturbative approach for thermal averages in quantum many-body systems: The thermal-cluster-cumulant method.

We present in this paper a systematic nonperturbative cluster-cumulant method for deriving averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to imaginary time. We have generalized the concepts of in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed cluster cumulants by us. For a compact treatment, we have derived an alternative field theory in which a time-ordered product is expanded in terms of thermal normal of operators and contractions-leading to a thermal Wick's theorem. The normal products are the finite-temperature analogs of the ordinary normal products and have zero averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch-Balian-de Dominicis theory.

Effects of anharmonicity on nonclassical states of the time-dependent harmonic oscillator.

Harmonic oscillators with time-dependent frequencies describe physics of the Paul trap of ions, radiation fields in media of time-varying dielectric constants, and so on. Their ground states are generically squeezed states. Here the effects of small anharmonicity on such nonclassical states are discussed. A perturbative calculation is performed, in which the unperturbed states are prepared in terms of the Hermitian invariant of the time-dependent harmonic oscillator. A quartic potential perturbation is investigated in detail. It is found that the correction term as well as the unperturbed term in the ground-state persistence probability can be expressed by the moments of a number operator alone. The oscillatory behavior is found in that correction term, which shows a competition of excitation and deexcitation due to the time dependence of the frequency and the nonlinear interaction. This may be observed experimentally as the temporal fluctuation of the quasienergy levels. Some formulas are also presented for the variances and correlations of the quadratures in the perturbed states.

Energy levels of a three-dimensional anharmonic oscillator with quartic anisotropic perturbation

Several energy levels are calculated for a three-dimensional oscillator with a quartic anisotropic perturbation, using a Hill determinant approach and an inner product perturbation technique, for a range of perturbation parameters. The results of the two methods are in good agreement.

Thermal averaging in quantum many-body systems: a non-perturbative thermal cluster cumulant approach

We present here a non-perturbative cluster cumulant approach for effecting thermal averaging in quantum-mechanical observables. We introduce a new representation of thermo field dynamics (TFD), with the associated thermal normal order and Wick expansion, and replace the thermal averaging by a quantum mechanical one involving a thermal “base state” |0 β〉. Unlike the traditional TFD, only the physical variables appear in our formulation. The imaginary time evolution of the statistical operator is treated by our recently developed time-dependent coupled cluster theory. The partition function Z is evaluated as a completely contracted quantity involving |0 β〉. We have computed Z for an anharmonic oscillator with quartic perturbation as an illustrative application. We also demonstrate numerically the validity of the Kohn—Luttinger theorem.

Anharmonic phonons and the isotope effect in superconductivity.

Anharmonic interionic potentials are examined in an Einstein model to study the unusual isotope-effect exponents for the high-${\mathit{T}}_{\mathit{c}}$ oxides. The mass dependences of the electron-phonon coupling constant \ensuremath{\lambda} and the average phonon frequency \ensuremath{\surd}〈${\mathrm{\ensuremath{\omega}}}^{2}$〉 are computed from weighted sums over the oscillator levels. The isotope-effect exponent is depressed below 1/2 by either a double-well potential or a potential with positive quadratic and quartic parts. Numerical solutions of Schr\"odinger's equation for double-well potentials produce \ensuremath{\lambda}'s in the range 1.5--4 for a material with a vanishing isotope-effect parameter \ensuremath{\alpha}. However, low phonon frequencies limit ${\mathit{T}}_{\mathit{c}}$ to roughly 15 K. A negative quartic perturbation to a harmonic well can increase \ensuremath{\alpha} above 1/2. In the extreme-strong-coupling limit, \ensuremath{\alpha} is 1/2, regardless of anharmonicity.

New quantum mechanical perturbation technique using an ‘‘electronic scratchpad’’ on an inexpensive computer

The authors have developed a new method for doing numerical quantum mechanical perturbation theory. It has the flavor of Rayleigh–Schrödinger perturbation theory (division of the Hamiltonian into an unperturbed Hamiltonian and a perturbing term, use of the basis formed by the eigenfunctions of the unperturbed Hamiltonian) while turning out to be a variational technique. Furthermore, it is easily implemented by means of the widely used ‘‘electronic scratchpad,’’ MathCAD 2.0, using an inexpensive computer. As an example of the method, the problem of a harmonic oscillator with a quartic perturbing term is examined.

Matrix-second order differential equations and chaotic Hamiltonian systems

We consider matrix-second order differential equations which are perturbations of the harmonic flow on the space of matrices. Experimental evidence of the non integrability of the two degrees of freedom Hamiltonian system provides an indication of the non existence of a Lax pair with commuting eigenvalues for perturbations of order six. This shows the specificity of quartic perturbations for which such a Lax pair was precedently obtained.

An analytic semiclassical model for local and normal stretching modes in AB 2 molecules

We present a simple, analytic model for the normal and local mode behavior of the stretching vibrations in AB 2 molecules. Starting with a zeroth-order description of degenerate, harmonic normal modes, we introduce a quartic perturbation and employ first-order secular perturbation theory to obtain a closed-form expression for the Poincaré surface of section for the action-angle variable pair corresponding to the vibrational angular momentum. Local modes correspond to pairs of levels trapped below a dynamical barrier along this angle variable; normal modes, to levels above the barrier. Uniform semiclassical quantization completely lifts the degeneracy of the zeroth-order states, and predicts the appropriate trends in the ordering of the various modes and the splitting between local mode pairs.

Exciton and vibronic effects in the spectroscopy of bianthracene in supersonic beams

Excitation and dispersed fluorescence spectra of 9,9′-bianthracene in a supersonic expansion are reported. The spectra are anthracene-like, indicating that the rings are weakly coupled. Exciton effects are considered in the interpretation of the spectra. The torsional potential in S1 is modeled as a double-well (Gaussian perturbation on a one-dimensional harmonic oscillator) with barriers to perpendicularity and planarity of ∼30 and ∼1100 cm−1, respectively. The S0 torsional potential shows negative anharmonicity which is modeled as a quartic perturbation. Anthracenic modes in S1 and S0 are also assigned. Finally, measurements of S1 fluorescence lifetimes up to ∼6000 cm−1 excess energy in the excited state show no evidence of charge transfer.

Perturbation series for transition moments of anharmonic oscillators

The off-diagonal hypervirial theorems and a matrix product rule are applied to the perturbed one-dimensional harmonic oscillator in a new method of obtaining perturbation series for the off-diagonal elements (m mod x mod n). The author gives the Rayleigh-Schrodinger series for (0 mod x mod 1) to eighth order for a quartic perturbation.