Unified Krylov-Bogoliubov-Mitropolskii method for solving nth order non-linear systems with slowly varying coefficients

The unified Krylov-Bogoliubov-Mitropolskii (KBM) method is extended for obtaining the transient response of an n-th order (n?2) non-linear system with slowly varying coefficients. The method is a generalization of KBM method and covers all the three cases when the eigenvalues of the unperturbed equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the eigenvalues in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.     

Long-time limit behavior of the solution to an atom’s master equation

We study the long-time limit behavior of the solution to an atom’s master equation.For the first time we derive that the probability of the atom being in the α-th(α = j + 1-j z,j is the angular momentum quantum number,j z is the z-component of angular momentum) state is {(1- K/G)/[1-(K/G) 2j+1 ]}(K/G) α-1 as t → +∞,which coincides with the fact that when K/G > 1,the larger the α is,the larger the probability of the atom being in the α-th state(the lower excited state) is.We also consider the case for some possible generalizations of the atomic master equation.     

EPR entangled states and complex fractional Fourier transformation

Starting from the Einstein-Podolsky-Rosen entangled state representations of continuous variables we derive a new formulation of complex fractional Fourier transformation (CFFT). We find that two-variable Hermite polynomials are just the eigenmodes of the CFFT. In this way the CFFT is linked to the appropriate operator transformation between two kinds of entangled states in the context of quantum mechanics. In so doing, the CFFT of quantum mechanical wave functions can be derived more directly and concisely. Received 21 February 2002 / Received in final form 1st June 2002 Published online 24 September 2002 RID="*" ID="*"Work supported by National Natural Science Foundation of China under grant 10175057 and the President Foundation of Chinese Academy of Science. RID="a" ID="a"e-mail: fhym@ustc.edu.cn     

Quantum graph vertices with permutation-symmetric scattering probabilities

Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix U. Observing that if U has at most two eigenvalues, then the scattering matrix S(k) of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta k, the transmission probability from the j-th edge to ?-th edge is independent of (j,?), and all the reflection probabilities are equal. We classify these couplings according to their scattering properties, which leads to the concept of generalized δ- and δ^′-couplings.     

应用特征光谱线性反演模型快速提取矿物的光谱预处理研究

矿物成分的野外快速鉴定在遥感地质研究与矿产勘探中至关重要,特征光谱线性反演模型可利用野外获得的岩矿光谱信息快速提取矿物组分,但是在实际使用中发现,使用同一样品的不同类型反射率光谱,模型结果存在显著差异.文章通过对连续统快速傅里叶变换(CFFT)滤波方法和特征光谱线性反演模型(CSLM)的研究,一方面获得了使用CFFT研究岩矿光谱的最佳参数设置,设置低通滤波截止频率为150Hz;另一方面通过对CSLM使用不同光谱数据结果的评价分析,发现使用CFFT滤波后的ASD光谱在野外提取矿物时效果更好.     

Numerical experiment of anharmonic oscillators by using the symplectic scheme-shooting method

Symplectic scheme-shooting method (SSSM) is applied to solve the energy eigenvalues of anharmonic oscillators characterized by the potentials V(x)=λx4 and V(x)=(1/2)x2+λx2α with α=2,3,4 and doubly anharmonic oscillators characterized by the potentials V(x)=(1/2)x2+λ1x4+λ2x6, and a high order symplectic scheme tailored to the "time"-dependent Hamiltonian function is presented. The numerical results illustrate that the energy eigenvalues of anharmonic oscillators with the symplectic scheme-shooting method are in good agreement with the numerical accurate ones obtained from the non-perturbative method by using an appropriately scaled basis for the expansion of each eigenfunction; and the energy eigenvalues of doubly anharmonic oscillators with the sympolectic scheme-shooting method are in good agreement with the exact ones and are better than the results obtained from the four-term asymptotic series. Therefore, the symplectic scheme-shooting method, which is very simple and is easy to grasp, is a good numerical algorithm.     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Bethe ansatz and boundary energy of the open spin-1/2 XXZ chain

We review recent results on the Bethe ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters α_?, α₊, β_?, β₊ are nonzero. A generalization of the BaxterT-Q equation that involves more than one independentQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.     

Use of anisotropic atom-atom potential functions in lattice-dynamical calculations for solid nitrogen

For solid nitrogen, a set of ‘anisotropic’ atom-atom semi-empirical potential functions, which are easily derived from the usual ‘6-exp’ or ‘12-6’, eliminates the difficulties encountered in demonstrating the stability of the γ-phase in lattice-dynamical calculations according to the Born-von Karman procedure.

Negative eigenvalues of the dynamical matrices in certain regions of the Brillouin zone disappear, the agreement with experimental data improves and the α-γ transition is foreseen.     

多模复共轭奇、偶相干态光场的N次方Y压缩与N次方H压缩

利用新近提出的多模辐射场的广义非线性等阶高阶压缩理论,对多模复共轭奇、偶相干态光场的N次方Y压缩与N次方H压缩特性进行了详细研究.结果发现:①多模复共轭奇相干态光场,压缩阶数N为偶数时,只存在N-Y最小测不准态;而当腔模总数q与压缩阶数的乘积q·N为偶数时,只存在N-H最小测不准态;此外,无论N及q·N为何值,不呈现N次方Y压缩与N次方H压缩效应.②多模复共轭偶相干态光场在一定条件下可呈现出周期性变化的任意阶的N次方Y压缩与N次方H压缩效应.③多模偶相干态光场与多模复共轭偶相干态光场两者的压缩效果和压缩特性完全相同,这种现象称为“压缩简并”.     

On long-time limit behavior of the solution of atom's master equation

We study long-time limit behavior of the solution of atom's master equation, for the first time we derive that the probability of the atom being in the α-th (α =j+1-j_z, j is the angular momentum quantum number, j_z is the z-component of angular momentum) state is {(1-K/G)/[1-(K/G)^2j+1]}(K/G)^α-1 as t→+∞, which coincides with the fact that when K/G > 1, the larger the α is, the larger probability of the atom being in the α-th state (the lower excited state). We also consider the case for some possible generalizations of the atomic master equation.     

Stability of the lepton-flavor mixing matrix against quantum corrections

Recent neutrino experiments suggest strong evidence of tiny neutrino masses and the lepton-flavor mixing. Neutrino-oscillation solutions for the atmospheric neutrino anomaly and the solar neutrino deficit can determine the texture of the neutrino mass matrix according to the neutrino mass hierarchies as Type A: , Type B: , and Type C: , where is the i-th generation neutrino mass. In this paper we study the stability of the lepton-flavor mixing matrix against quantum corrections for all three types of mass hierarchy in the minimal supersymmetric Standard Model with an effective dimension-five operator which gives the Majorana masses of neutrinos. The relative sign assignments of neutrino masses in each type play crucial role for the stability against quantum corrections. We find that the lepton-flavor mixing matrix of Type A is stable against quantum corrections, and that of Type B with the same (opposite) signs of and are unstable (stable). For Type C, the lepton-flavor-mixing matrix approaches the definite unitary matrix according to the relative sign assignments of the neutrino mass eigenvalues as the effects of quantum corrections become large enough to neglect the squared mass differences of neutrinos. Received: 24 June 1999 / Revised version: 23 December 1999 / Published online: 17 March 2000     

一种新型的多模虚偶相干态光场的等阶N次方Y压缩与等阶N次方H压缩

本文利用新近建立的多模压缩态理论,详细研究了一种新型的多模虚偶相干态光场|Ψ_i,e⁽²⁾>_q的广义非线性等阶N次方Y压缩与等阶N次方H压缩特性.结果发现:1)态|Ψ_i,e⁽²⁾>_q是一种典型的多模非经典光场,当压缩阶数N为奇数时,态|Ψ_i,e⁽²⁾>_q在一定条件下总可呈现出周期性变化的、任意阶的等阶N次方Y压缩效应;当腔模总数q与压缩阶数N这两者的乘积q·N为奇数时,则在一定条件下态|Ψ_i,e⁽²⁾>_q又可呈现出周期性变化的、任意阶的等阶N次方H压缩效应.2)态|Ψ_i,e⁽²⁾>_q的等阶N次方Y压缩与等阶N次方H压缩效应这两者的压缩程度和压缩深度分别与几率幅γ_q^(e)、压缩参数R_j、各模的初始相位φ_j(或者各模的初始相位和 φ_j)、压缩阶数N以及腔模(指纵模)总数q等呈较强的非线性关联,等阶N次方H压缩效应与上述诸参量之间的非线性关联程度要比等阶N次方Y压缩效应的更强.3)多模虚偶相干态光场|Ψ_i,e⁽²⁾>_q与多模偶相干态光场|Ψ_,e>_q及多模复共轭偶相干态光场|Ψ*_,e⁽²⁾>_q这后两者的等阶N次方Y压缩效应和等阶N次方H压缩效应的压缩条件和压缩特性正好相反,这种现象就称为相反压缩.     

Continuous Particles in the Canonical Ensemble as an Abstract Polymer Gas

We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of N continuous particles in the Canonical Ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the Grand Canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the free energy can be written as a series in powers of the density whose k-th order coefficient coincides, modulo terms o(N)/N, with the k-th order virial coefficient divided by k+1, according to its expression in terms of the m-th order (with m≤k+1) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the k-th order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.     

一类一维势场中电子态的研究

研究了在一维势场Ｖ_n＝λｃｏｓ（Ｑn＋ａｎ^ｖ）（０＜ｖ＜１）中运动的电子状态，计算了本征能量和本征态的局域化指数。对Ｑ＝２π／３，系统的能带由三个子能带构成。当λ小于２时，每个子带中有两个迁移率边界。研究了扩展态、局域态以及迁移率边界随参数λ，ν，α的变化。 关键词：     

A linear relationship exists among brain diffusion eigenvalues measured by diffusion tensor magnetic resonance imaging

Diffusion in biological tissues can be measured by magnetic resonance diffusion tensor imaging The complex nature of anisotropic diffusion in the brain has been described by a diffusion tensor which contains information about the magnitude of diffusion in different directions. Each tensor contains a set of three eigenvalues which are related to the major, intermediate, and minor axes of a diffusion ellipsoid. This investigation demonstrates that the various sets of diffusion eigenvalues from different regions of the brain lie along a line in ordered eigenvalue space. Sets of ordered diffusion eigenvalues were considered points in ordered eigenvalue space. The line which best fit the data by minimizing the total squared deviations was determined. A new coordinate system was constructed through translation and rotation which spanned ordered eigenvalue space. Eigenvalues from both monkey brain and human brain were studied. It was found that the sets of eigenvalues from both species have significant linear trends. Moreover, the same line may describe the brain eigenvalues from both species. It is likely that this linear relationship of the eigenvalues observed in an ordered eigenvalue plot is related to a combination of (1) conservation of total isotropic diffusion and (2) the degree of orientational dispersion of the microfibers within each voxel.     

Three-dimensional dissipative optical solitons

A brief overview of recent theoretical results in the area of three-dimensional dissipative optical solitons is given. A systematic analysis demonstrates the existence and stability of both fundamental (spinless) and spinning three-dimensional dissipative solitons in both normal and anomalous group-velocity regimes. Direct numerical simulations of the evolution of stationary solitons of the three-dimensional cubic-quintic Ginzburg-Landau equation show full agreement with the predictions based on computation of the instability eigenvalues from the linearized equations for small perturbations. It is shown that the diffusivity in the transverse plane is necessary for the stability of vortex solitons against azimuthal perturbations, while fundamental (zero-vorticity) solitons may be stable in the absence of diffusivity. It has also been found that, at values of the nonlinear gain above the upper border of the soliton existence domain, the three-dimensional dissipative solitons either develop intrinsic pulsations or start to expand in the temporal (longitudinal) direction keeping their structure in the transverse spatial plane. Presented at 9-th International Workshop on Nonlinear Optics Applications, NOA 2007, May 17–20, 2007, Świnoujście, Poland     

Nichtlineare Wechselwirkung elektromagnetischer Wellen im Plasma mit konstantem Magnetfeld

The nonlinear interaction of waves in anisotropic plasmas is considered. Solving the hydrodynamics equations with a n-th order perturbation procedure, an equation for the nonlinear electric field is derived. The n-th order current produced by interaction of waves with lower order is represented by means of a nonlinear tensor of conductivity, which is calculated for n = 2. It is shown, that a miced plasma wave with combination frequencies is excited as the result of interaction of two transverse (ordinary) waves propagating perpendicular to the magnetic field.     

Adiabatic approximation in PT-symmetric quantum mechanics

In this paper, we present a quantitative sufficient condition for adiabatic approximation in PT-symmetric quantum mechanics, which yields that a state of the PT-symmetric quantum system at any time will remain approximately in the m-th eigenstate up to a multiplicative phase factor whenever it is initially in the m-th eigenstate of the Hamiltonian. In addition, we estimate the approximation errors by the distance and the fidelity between the exact solution and the adiabatic approximate solution to the time evolution equation, respectively.     

Courant-Sharp Eigenvalues for the Equilateral Torus,and for the Equilateral Triangle

We address the question of determining the eigenvalues \({\lambda_{n}}\) (listed in nondecreasing order, with multiplicities) for which Courant’s nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with \({n}\) nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.