Solving Energy Levels for SSH Hamiltonian Describing Peierls Phase Transition by Virtue of Invariant Eigen-operator Method

We show that the recently proposed invariant eigen-operator （IEO） method can be successfully applied to solving energy levels for SSH Hamiltonian describing Peierls phase transition. The electronic energy band of compound lattice is also studied by IEO method.

Applying invariant eigen-operator method to deriving normal coordinates of general classical Hamiltonian

For classical Hamiltonian with general form H = 1/2∑ijMijpipj＋1/2∑ijLijqiqj we find a new convenient way to obtain its normal coordinates, namely, let H be quantised and then employ the invariant eigen-operator （IEO） method （Fan et al. 2004 Phys. Lett. A 321 75） to derive them. The general matrix equation, which relies on M and L, for obtaining the normal coordinates of H is derived.

Deriving Energy-Gap of Some Nonlinear Hamiltonians by Invariant Eigen-operator Method

By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for someHamiltonians describing nonlinear processes in particle physics.In this way the energy-gap of the Hamiltonians can benaturally obtained.The characteristic polynomial theory has been fully employed in our derivation.

Deriving Energy-Level Gap of a System in Presence of a Kerr-like Medium by Virtue of Invariant Eigen-operator Method

In this paper, we find the invariant eigen-operators （IEOs） and the energy-level gap of a system with a two-level atom interacting with single mode cavity field through multi-photon transition in the presence of a Kerr-like medium. From this work, one can see that the IEO method in many eases is simpler and easier on obtaining the energy-level gap formula than the usual way.

Energy Gap for Some Hamiltonian Systems Describing Polariton Systems Obtained via Invariant Eigen-operator Method

Based on the invariant eigen-operator method （lEO） [Phys. Left. A 321 （2004） 75] we derive the exact energy gap for some Hamiltonians, which describe some polariton systems. The result shows that in some cases the IEO method, stemming from the Heisenberg approach, is more direct and convenient for deriving the energy-level gap formula than via the approach of solving the Schrodinger equation.

Pseudo-invariant Eigen-operator for Deriving Energy-Level Gap for Jaynes-Cummings Model

We extend the concept of invariant eigen-operator to pseudo-invariant eigen-operator case through analyzing the standard Jaynes-Cummings model. We find the pseudo-invariant eigen-operator in terms of supersymmetric generators of this model, which diretly leads to the energy-level gap for Jaynes Cummings Hamiltonian.

Abundant invariant solutions of extended(3+1)-dimensional KP-Boussinesq equation

Lie group analysis method is applied to the extended(3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation and the corresponding similarity reduction equations are obtained with various infinitesimal generators.By selecting suitable arbitrary functions in the similarity reduction solutions,we obtain abundant invariant solutions,including the trigonometric solution,the kink-lump interaction solution,the interaction solution between lump wave and triangular periodic wave,the two-kink solution,the lump solution,the interaction between a lump and two-kink and the periodic lump solution in different planes.These exact solutions are also given graphically to show the detailed structures of this high dimensional integrable system.

Normal coordinate in harmonic crystal obtained by virtue of the classical correspondence of the invariant eigen-operator

Noticing that the equation with double-Poisson bracket, where On is normal coordinate, Hc is classical Hamiltonian, is the classical correspondence of the invariant eigen-operator equation （2004 Phys. Left. A. 321 75）, we can find normal coordinates in harmonic crystal by virtue of the invaxiant eigen-operator method.

Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations

An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving non-linear hyperbolic systems is developed.The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements.The method is made invar-iant domain preserving for the Euler equations using convex limiting and is tested on vari-ous benchmarks.

Energy level of an electron in a saddle-potential quantum dot under a uniform magnetic field obtained by the invariant eigenoperator method

We show that the recently proposed invariant eigenoperator method can be successfully applied to solving the energy levels of an electron in a saddle-potential quantum dot under a uniform magnetic field. The Landau diamagnetism decreases with the value wy2 - wx2 due to the existence of the saddle potential.

A fast-imaging method for airborne gravity gradient data based on tensor invariants

Airborne gravity gradient data contain additional short-wavelength information about the buried geological bodies.This study develops a fast interpretation method based on the gravity gradient data for the sources’spatial location and physical property parameters.This study analyzes the advantages of the source parameter inversion method based on tensor invariants.It proposes a normalized fast-imaging method based on tensor invariants to quickly estimate the spatial location parameters of sources through the local maximum value position of the imaging results.First,the tensor invariant characteristics and the imaging method’s effect in a simple model are analyzed using a theoretical model.Second,to analyze the imaging method’s application effect in complex model conditions,the method’s applicability is quantitatively analyzed using the data added with noise,superimposed anomalies of adjacent sources,and anomalies of deep and shallow geological bodies.The theoretical model’s simulation results show that the model’s imaging results in this study have satisfactory performance on the spatial position estimation of the sources.Finally,the method is applied to the gravity anomaly data corresponding to the Humble salt dome.The imaging results can effectively estimate the distribution of the salt dome’s horizontal and depths,verifying the practicability of the method.

An iterative Wiener filtering method based on the gravity gradient invariants

How to deal with colored noises of GOCE （Gravity field and steady - state Ocean Circulation Explorer） satellite has been the key to data processing. This paper focused on colored noises of GOCE gradient data and the frequency spectrum analysis. According to the analysis results, gravity field model of the optima] degrees 90-240 is given, which is recovered by COCE gradient data. This paper presents an iterative Wiener filtering method based on the gravity gradient invariants. By this method a degree-220 model was calculated from GOCE SGG （Satellite Gravity Gradient） data. The degrees above 90 of ITG2010 were taken as the prior gravity field model, replacing the low degree gravity field model calculated by GOCE orbit data. GOCE gradient colored noises was processed by Wiener filtering. Finally by Wiener filtering iterative calculation, the gravity field model was restored by space-wise harmonic analysis method. The results show that the model＇s accuracy matched well with the ESA＇s （European Space Agency） results by using the same data,

Generation and classification of the translational shape-invariant potentials based on the analytical transfer matrix method

For the conventional translational shape-invariant potentials （TSIPs）, it has demonstrated that the phase contribution devoted by the scattered subwaves in the analytical transfer matrix quantization condition is integrable and independent of n. Based on this fact we propose a novel strategy to generate the whole set of conventional TSIPs and classify them into three types. The generating functions are given explicitly and the Morse potential is taken as an example to illustrate this strategy.

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators.

Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full. description, of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invaxiant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite

Formal Calculation and Invariant-Based Validation Establish Dependable Algorithmic Programs

The paper presents a formal and practical approach to dependable algorithm development.First,starting from a formal specification based on the Eindhoven quantifier notation,a problem is regularly reduced to subproblems with less complexity by using a concise set of calculation rules,the result of which establishes a recurrence-based algorithm.Second,a loop invariant is derived from the problem specification and recurrence,which certifies the transformation from the recurrence-based algorithm to one or more iterative programs.We demonstrate that our approach covers a number of classical algorithm design tactics,develops algorithmic programs together with their proof of correctness,and thus contributes fundamentally to the dependability of computer software.

PERSISTENCE OF RESONANT INVARIANT TORI WITH NON-HAMILTONIAN PERTURBATION

A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone’s for the three dimensional non-Hamiltonian systems.

The invariant of the stiffness filter function with the weight filter function of the power function form

Based on the independent, continuous and map- ping （ICM） method and homogenization method, a research model is constructed to propose and deduce a theorem and corollary from the invariant between the weight filter func- tion and the corresponding stiffness filter function of the form of power function. The efficiency in searching for op- timum solution will be raised via the choice of rational filter functions, so the above mentioned results are very important to the further study of structural topology optimization.

Invariant expression for tunneling rate under canonical transformations

The original formula to calculate the tunneling rate through event horizons is apparently dependent on the type of coordinates used. In this paper, we propose an invariant expression under canonical transformations to study the tunneling effect. Moreover, the problem of factor 2 is solved naturally. As an application of this expression, we obtain the same tunneling rate both in the Schwarzschild and the Painlev6 coordinates. It is shown that once the suitable formula to calculate tunneling rate is correctly identified, the tunneling method is manifestly covariant.

Group Invariant Solutions of the Two-Dimensional Elastodynamics Problem in the Polar Coordinate System

Some group invariant solutions of the two-dimensional elastodynamics problem in linear homogeneous isotropic materials are considered using the group-theoretical method.In the polar coordinate system,three group invariant solutions are constructed by the invariants of the Lie group,which are admitted by the governing equations for the two-dimensional elastodynamics problem.The graphical figures of the group invariant solutions are presented,and the physical meanings of the group invariant solutions are expounded in some cases.