A Note on Unification of Translational Shape Invariant Potential and Scaling Shape Invariant Potential

This article puts forward a general shape invariant potential, which includes the translational shape invariant potential and scaling shape invariant potential as two particular cases, and derives the set of linear differential equations for obtaining general solutions of the generalized shape invariance condition.     

精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

超对称WKB近似与一维无限深势阱

证明了一维无限深势阱为非形状不变势.采用超对称WKB近似,得到了一维无限深势阱的精确能谱公式.计算结果表明,把势场具有形状不变性作为超对称WKB近似能够给出精确能谱的充要条件,是一个过强的提法,需要加以弱化.     

Matched Spatial Filtering by Primitive Reference Patterns and Correlation Diagrams for Shape Invariant Pattern Recognition

In this paper, twelve linear primitive reference patterns are proposed for synthesizing a matched spatial filter, known as a shape invariant filter, which can recognize all patterns with various shapes. Furthermore, reprocessed data, given by correlation between the input object of the system and the reference patterns “correlation diagrams” are introduced for processing peak signals in a digital system. By correlating the diagram from the input object with the standard diagrams, the 26 of the alphabet letters were recognized in the experiment. Moreover, feasibility of the rotational invariance of the filter was suggested by the computer simulation.     

Spectral properties of supersymmetric shape invariant potentials

We present the spectral properties of supersymmetric shape invariant potentials (SIPs). Although the folded spectrum is completely random, unfolded spectrum shows that energy levels are highly correlated and absolutely rigid. All the SIPs exhibit harmonic oscillator-type spectral statistics in the unfolded spectrum. We conjecture that this is the reflection of shape invariant symmetry.      

On New Invariant Solutions of Generalized Fokker-Planck Equation

The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first reported.     

Shape invariant potentials in higher dimensions

In this paper we investigate the shape invariance property of a potential in one dimension. We show that a simple ansatz allows us to reconstruct all the known shape invariant potentials in one dimension. This ansatz can be easily extended to arrive at a large class of new shape invariant potentials in arbitrary dimensions. A reformulation of the shape invariance property and possible generalizations are proposed. These may lead to an important extension of the shape invariance property to Hamiltonians that are related to standard potential problems via space time transformations, which are found useful in path integral formulation of quantum mechanics.     

Six—Parameter Exponential—Type Potential and the Identity for the Exponential—Type Potentials

We propose a six-parameter exponential-type potential (SPEP),which has been shown to be a shapeinvariant potential with a translation of parameters.For this reducible potential,the exact energy levels are obtained by using the supersymmetric shape invariance technique.Choosing appropriate parameters,four classes of exponential-type potentials and their exact energy spectra are reduced from the SPEP and a general energy level formula,respectively.Each class shows the identity except for the different definitions of parameters.     

Mapping of Shape Invariant Potentials by the Point Canonical Transformation

In this paper by using the method of point canonical transformation we find that the Coulomb and Kratzer potentials can be mapped to the Morse potential. Then we show that the Pöschl-Teller potential type I belongs to the same subclass of shape invariant potentials as Hulthén potential. Also we show that the shape-invariant algebra for Coulomb, Kratzer, and Morse potentials is SU(1,1), while the shape-invariant algebra for Pöschl-Teller type I and Hulthén is SU(2).     

用不变本征算符法求晶面吸附原子的振动模

晶体表面的扩散和缺陷对晶体振动模式的影响是表面物理学研究的一个重要和基本的课题.晶格振动的频率对应于系统的能带.由于晶格中原子的振动不是孤立的,并且晶格具有周期性,所以在晶体中形成格波.格波代表晶体中所有原子都参与的频率相同的振动,又常称为一种振动模.本文讨论在表面吸附位势系数β_0与晶体内部原子的周期位势系数β不同的情况下,晶体表面吸附一个质量为m_0(与晶格原子质量m不同)的原子以后晶格的振动模.采用不变本征算符方法,严格地导出此振动模为ω=((2β(1-coshα))/(hm))~(1/2),其中α=ln[-(mβ_0+m_0(-2β+β_0)+(β_0)~(1/2)((-4mm_0β+(m+m_0)~2β_0))~(1/2)/2m_0β].此结果表明,ω不但取决于吸附位势与吸附原子的质量,也与晶格原子的质量与内位势有关.     

Identity for the Exponential-Type Molecule Potentials and the Supersymmetry Shape Invariance

The identity and the supersymmetry shape invariance for a class of exponential-type molecule potentials are studied by introducing a deformed five-parameter exponential-type potential (DFPEP) and via the multi-parameter deformations. It has been shown that the DFPEP is a shape-invariant potential with a translation of parameters. By making use of the shape invariance approach, the exact energy levels are determined for the bound states with zero angular momentum. A class of molecule potentials and their exact energy spectra for the zero angular momentum states are reduced from the DFPEP and a general energy spectrum formula, respectively. The interrelations for some molecule potentials are also discussed.     

Rational solutions for the Riccati-Schrödinger equations associated to translationally shape invariant potentials

We develop a new approach to build the eigenfunctions of a translationally shape invariant potential. For this we show that their logarithmic derivatives can be expressed as terminating continued fractions in an appropriate variable. We give explicit formulas for all the eigenstates, their specific form depending on the Barclay-Maxwell class to which the considered potential belongs.     

Invariant manifolds associated to nonresonant spectral subspaces

We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain nonresonance conditions, the map leaves invariant a smooth manifold tangent to this subspace. This manifold is as smooth as the map—when the smoothness is measured in appropriate scales—but is unique amongC ^L invariant manifolds, whereL depends only on the spectrum of the linearization or on some more general smoothness classes that we detail. We show that if the nonresonance conditions are not satisfied, a smooth invariant manifold need not exist, and we also establish smooth dependence on parameters. We also discuss some applications of these invariant manifolds and briefly survey related work.     

Point Canonical Transformation for Solving Five-Parameter Exponential-Type Potential

Using the approach of mapping of shape invariant potentials under point canonical transformations, the energy spectra and wave functions are most easily determined for the bound states of the five-parameter exponential-type potential with a little extra effort.     

用不变量方法解具有非线性场和原子与场非线性耦合的Jaynes-Cummings模型

采用Lewis-Riesenfeld不变量方法研究了具有非线性场和任意形式原子与场相互作用的Jaynes-Cummings模型.该模型由于具有超对称代数结构,因此其Hamiltonian量可用超对称算符的线性组合表示.在算符N′的本征值子空间,用生成元算符构造出系统的不变量后,利用不变量方法求得了系统的一般波函数和时间演化算符,同时也计算了原子布居数和光子数的时间演化表达式.     

Knotted Wave Dislocation with the Hopf Invariant

We study the wave dislocations with an induced gauge potential. The topological current characterized the wave dislocations is constructed with the dual of Abelian gauge field. And the topological charges and locations of the wave dislocations are determined by the φmapping topological current theory. Furthermore, it is shown that the knotted wave dislocations can be described with a Hopf invariant in the wave field. At last we discussed the evolution of the knotted wave dislocations. PACS 02.10.Kn, 02.40.-k, 11.15.-q     

The scattering amplitude for a newly found exactly solvable potential

The scattering amplitude for a recently discovered exactly solvable shape invariant potential, which is isospectral to the generalized Pöschl–Teller potential, is calculated explicitly by considering the asymptotic behavior of the X₁$X_1$ Jacobi exceptional polynomials associated with this system.     

基于线性不变矩的特征图象配准算法研究

针对图象差异较大的光学图象与合成孔径雷达图象(SAR)的特点,设计了一种基于特征的图象配准的算法,包括特征提取、匹配、控制点选取、变换系数计算和误差计算等步骤.在特征提取中,对于雷达图象需要进行预处理并采用LOG提取图象轮廓.在匹配算法中,以线性不变矩作为特征量,对现有基于图象轮廓特征的匹配算法进行了改进.采用上海浦东地区的Radarsat-1S2模式图象与LandsatTM-5波段图象作为待配准图象.实验结果表明,该方法可以较好地完成光学图象与SAR图象的配准,有着较高的应用价值.     

Relativity Groups for Spacetimes with Invariant Velocities

New relativity groups for spacetime admitting invariant velocities are considered. It is shown that most of them restrict the maximal allowed velocities of reference frames.     

Supersymmetry and Shape Invariance of Hartmann Potential and Ring-Shaped Oscillator Potential in the γ and θ Dimensions of Spherical Polar Coordinates

This article shows that in spherical polar coordinates, some noncentral separable potentials have super-symmetry and shape invariance in the r and θ dimensions, we choose Hartmann potential and ring-shaped oscillator astwo important examples, thus in principle the energy eigenvalues and energy eigenfunctions of such the potentials in ther and θ dimensions can be obtained by the method of supersymmetric quantum mechanics. Here we use an alternativemethod to get the required results.