New q-Deformed Oscillator Algebra and Thermodynamic Model

In this paper we propose the new q-oscillator algebra. We discuss the coherent state and the deformed su(2) algebra for this algebra when q is real. As is different from Arik–Coon algebra (J. Math. Phys. 17:524, 1976), this algebra is invariant under the hermitian conjugation for complex q. When q is a root of unity, we obtain the finite dimensional Fock space. Finally we discuss the thermodynamics of particle obeying this algebra when q is a root of unity.     

在FBF背景下带反射边界条件的超对称t-J模型的代数Bethe ansatz方法

在阶化量子反散射的框架中,得到FBF背景下,带反射边界条件的超对称t－J模型的本征值和本征矢,及相应的Betheansatz方程．     

N = 4 Supersymmetric Morse Oscillator and Its Spectrum-Generating Algebra

In this paper N = 4 supersymmetry of generalized Morse oscillators in one dimension is studied. Both bound states and scattering states of its four superpartner Hamiltonians are analyzed by using unitary irreducible representations of the noncompact Lie algebra su（1,1）. The spectrum-generating algebra governing the Hamiltonian of the N = 4 supersymmetric Morse oscillator is shown to be connected with the realization of Lie superalgebra osp（1,2） or B（0,1） in terms of the variables of a supersymmetric two-dimensional harmonic oscillator.     

Generalized Bessel Functions and Lie Algebra Representation

The generalized Bessel functions (GBF) are framed within the context of the representation Q(ω,m ₀) of the three-dimensional Lie algebra . The analysis has been carried out by generalizing the formalism relevant to Bessel functions. New generating relations and identities involving various forms of GBF are obtained. Certain known results are also mentioned as special cases.Mathematics Subject Classifications (2000) 33C10, 33C80, 33E20.     

N = 4 Supersymmetric Morse Oscillator and Its Spectrum-Generating Algebra

In this paper N = 4 supersymmetry of generalized Morse oscillators in one dimension is studied. Both bound states and scattering states of its four superpartner Hamiltonians are analyzed by using unitary irreducible representations of the noncompact Lie algebra su(1,1). The spectrum-generating algebra governing the Hamiltonian of the N = 4 supersymmetric Morse oscillator is shown to be connected with the realization of Lie superalgebra osp(1,2)or B(0,1) in terms of the variables of a supersymmetric two-dimensional harmonic oscillator.     

广义非简谐振子的多尺度微扰理论

应用多尺度微扰理论到广义非简谐振子, 得到了一阶经典和量子微扰解. 特别是 我们的量子解在极限条件下能方便地转变为经典解, 并且坐标和动量算符的对易 关系的简化十分自然. 与Taylor级数解相比较, 无论是在经典还是在量子解 中频率移动都出现在各阶振动表达式中, 所以多尺度微扰解是弱耦合非简谐振动的较好解法.     

Generalized Uncertainty Relation of One-Dimensional Rindler Oscillator

General Minkowski vacuum state is seen to be equivalent to a thermal bath for a Rindler uniformly accelerated observer. This paper calculates the generalized uncertainty relation of one-dimensional Rindler oscillator in the coordinate representation. The calculations show that for a Rindler uniformly accelerated observer there is not only general quantum fluctuation but also thermal fluctuation related to his acceleration.     

Generalized Spiked Harmonic Oscillator in Non-commutative Space

In this paper non-commutative Schrodinger equation is considered for generalized Spiked harmonic oscillator potential. The energy shift due to non-commutativeity is obtained via the perturbation theory. Furthermore we show that the degeneracy of the initial spectral line is broken in transition from commutative space to non-commutative space.     

贴片晶振在冲击环境下的损伤边界

贴片式石英晶体振荡器广泛应用于各类电子和通信设备系统中。针对晶振在冲击环境中容易出现结构破坏而导致系统工作异常的问题,通过分析单自由度系统在不同频率冲击载荷作用下的响应特点,建立了结构的应力响应水平与相关冲击响应谱谱值之间的联系,获得了较已有结论更合理的损伤边界形式。根据典型晶振结构易损组件的力学特性建立对应的简化分析模型,得到了贴片晶振在大频率范围内的结构损伤边界。利用有限元仿真软件,对晶振结构在0.5~30 kHz频率范围内冲击载荷下的响应进行仿真分析,验证了该损伤边界的有效性。这也为以贴片晶振为代表的微小元器件在冲击环境下的可靠性研究提供了一种可行的方法。     

Unitary Operator of su q (n)-Covariant Oscillator Algebra

The unitary operator of su _q (n)-covariant oscillator algebra is constructed and two types of q-coherent states are obtained explicitly.     

Generalized Harmonic Oscillator and the SchrSdinger Equation with Position-Dependent Mass

We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties corresponding effective potentials for several mass functions, for the system with PDM are also discussed. We give the the systems with such potentials are isospectral to the usual harmonic oscillator.     

Asymptotics of a Discrete-Time Particle System Near a Reflecting Boundary

We examine a discrete-time Markovian particle system on ?×?₊ introduced in Defosseux (arXiv:1012.0117v1). The boundary {0}×?₊ acts as a reflecting wall. The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall universality class. After projecting to a single horizontal level, we take the long-time asymptotics and obtain the discrete Jacobi and symmetric Pearcey kernels. This is achieved by showing that the particle system is identical to a Markov chain arising from representations of O(∞) (introduced in Borodin and Kuan (Commun. Pure. Appl. Math. 63(7):831–894, 2010, arXiv:0904.2607)). The fixed-time marginals of this Markov chain are known to be determinantal point processes, allowing us to take the limit of the correlation kernel. We also give a simple example which shows that in the multi-level case, the particle system and the Markov chain evolve differently.     

Classical Oscillator Model on Canonical, Lie-Algebraic Deformation and Heisenberg-Weyl Algebra Deformation Nonrelativistic Space

The oscillator model on nonrelativistic Canonical (soft), Lie-Algebraic deformation noncommutative space and deformed Heisenberg-Weyl Algebra noncomutative phase space are analyzed. For canonical deformation the additional dynamical effects are absent. For two kinds of Lie-Algebraic deformation space the additional effects are generated. Remarkably angular frequency ω _ρ is stepfunction, non-periodic function, and contains imaginary frequency. The corresponding particle trajectory varies greatly along with the deformation and time parameter $\hat{k}$ , t. For deformed Heisenberg-Weyl Algebra noncommutative phase space the additional corrections are generated. K _ρ ,ω _ρ and particle trajectory have constant second-order correction with deformation parameters Θ and Π, but the particle still keep trajectory of classical oscillator.     

A Type of New Loop Algebra and a Generalized Tu Formula

A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.     

Semiclassical Limits of Ore Extensions and a Poisson Generalized Weyl Algebra

We observe [Launois and Lecoutre, Trans. Am. Math. Soc. 368:755–785, 2016, Proposition 4.1] that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra A₁, considered as a Poisson version of the quantum generalized Weyl algebra, is constructed and its Poisson structures are studied. In particular, a necessary and sufficient condition is obtained, such that A₁ is Poisson simple and established that the Poisson endomorphisms of A₁ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.     

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger–Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra \({\mathfrak{u}(2)}\) . Two of the symmetry generators, J ₃ and J ₂, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J ₃ is diagonal and the operator J ₂ acts in a tridiagonal fashion. In the circular basis, the operator J ₂ is block upper-triangular with all blocks 2 × 2 and the operator J ₃ acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J ₂ in the circular basis are generated by the Heun polynomials, and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J ₂ are generated by little ?1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J ₂ is diagonal is considered. In this basis, the defining relations of the Schwinger–Dunkl algebra imply that J ₃ acts in a block tridiagonal fashion with all blocks 2 × 2. The matrix elements of J ₃ in this basis are given explicitly.