Asymptotic Energy Expansion for Rational Power Polynomial Potentials

Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form V(x)= x2n/m+b1xn1/m1+b2x n2/m2+ …+ bNxnN/mN where n, m, n1, m1, . . . , nN, mN are positive integers while coefficients bk∈C. As in the case of even degree polynomial potentials with integer powers, all the integrals in the expansion can be evaluated analytically in terms of Γ functions. With the help of two examples, we demonstrate the usefulness of these expansions in getting analytic insight into the quantum systems having rational power polynomial potentials.     

Asymptotic Energy Expansion for Rational Power Polynomial Potentials

Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form V(x)=x^2n/m+b₁x^n1/m1+b₂x^n2/m2 +··· + b_Nx^n_N/m_N where n,m,n₁,m₁,...,n_N,m_N are positive integers while coefficients b_k∈ C. As in the case of even degree polynomial potentials with integer powers, all the integrals in the expansion can be evaluated analytically in terms of Γ functions. With the help of two examples, we demonstrate the usefulness of these expansions in getting analytic insight into the quantum systems having rational power polynomial potentials.     

Asymptotic Lower Bound for the Relative Disparities of Truncated-Path-Integral Partition Functions

Any truncated-path-integral partition function of a nonrelativistic quantum system in thermodynamic equilibrium—one obtained by means of the Feynman path-integral-procedure using a finite number of such integrals—is known to have a value not less than that of the exact one corresponding to it. A rigorous asymptotic lower bound obtained for the relative disparity in their values—the difference in their values divided by that of the exact partition function— confirms asymptotic positive-definiteness of the original upper bound. Values determined directly for a linear harmonic oscillator agree asymptotically with values of they bound.     

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.     

Metaplectic Representation of ospq(1/2) Algebra and Basic Hypergeometric Functions

In this paper we use the bosonic realization of osp_q(1/2) algebra to obtain its metaplectic representation. The group element for this algebra is shown to be described in terms of the basic hypergeometric function.     

Some Questions of the Nevanlinna Theory for the Complex Half-Plane

In this paper we give variants of the logarithmic derivative lemma and the second main theorem for the complex half-plane. Here the second main theorem is a special case of a theorem concerning polynomial functions of the coordinates of a holomorphic curve f : C ₊ P ⁿ . This theorem is a variant for C ₊ of a theorem of A. Eremenko and M. Sodin, who considered an entire curve f : C ₊ P ⁿ . The proofs are realized because of the introduction of natural classes of subharmonic, -subharmonic, holomorphic, meromorphic functions in C ₊ . Such functions we shall call just functions.     

用一个特殊的表示空间计算Sn群[n—1,1]表示的实正交形式

利用一个特殊的表示空间, 给出了S_n群表示[n—1, 1]的实正交形式的一般公式. 并举例说明了该技巧的一个应用.     

动量表象中氢原子径向函数的完备集

利用高级超越函数的一些性质,得到了超几何函数的一个展开关系式,得到了动量表象中氢原子径向波函数的完备集。     

Construction of a Weyl Representation from a Weak Weyl Representation of the Canonical Commutation Relation

Weak Weyl representations of the canonical commutation relation (CCR) with one degree of freedom are considered in relation to the theory of time operator in quantum mechanics. It is proven that there exists a general structure through which a weak Weyl representation can be constructed from a given weak Weyl representation. As a corollary, it is shown that a Weyl representation of the CCR can be constructed from a weak Weyl representation which satisfies some additional property. Some examples are given. The work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Some New Properties of Wigner Function Studied in Entangled State Representation

Based on the Wigner operator in the entangled state representation we study some new important properties of Wigner function for bipartite entangled systems, such as size of an entangled state, upper bound of Wigner functions, etc. These discussions demonstrate the beauty and elegance of the entangled state representation.     

Asymptotic normality in a two-dimensional random walk model for cell motility

We prove the asymptotic normality of a two-dimensional random walk describing the locomotion of cells on planar surfaces and calculate the asymptotic covariance matrix. The trajectories of the walk are random broken lines covered with constant speed, where the time intervals between turns as well as the turn angles are random and stochastically independent.     

Relationship Between Wave Function and Corresponding Wigner Function Studied in Entangled State Representation

By using the explicit form of the entangled Wigner operator and the entangled state representation we derive the relationship between wave function and corresponding Wigner function for bipartite entangled systems. The technique of integration within an ordered product （IWOP） of operators is employed in our discussions.     

Correct Path-Integral Formulation of Quantum Thermal Field Theory in Coherent State Representation

The path-integral quantization of thermal scalar, vector, and spinor fields is performed newly in the coherent-state representation. In doing this, we choose the thermal electrodynamics and φ⁴ theory as examples. By this quantization, correct expressions of the partition functions and the generating functionals for the quantum thermal electrodynamics and φ⁴ theory are obtained in the coherent-state representation. These expressions allow us to perform analytical calculations of the partition functions and generating functionals and therefore are useful in practical applications. Especially, the perturbative expansions of the generating functionals are derived specifically by virtue of the stationary-phase method. The generating functionals formulated in the position space are re-derived from the ones given in the coherent-state representation.     

Phase Properties of Two-Mode Squeezing-Rotating Entangled Representation

By virtue of the squeezing-rotating entangled representation, we mainly establish the new two-mode phase operator and phase angle operat, or, which is a general form including the foregoing formalist in two-mode Fock space. In addition, the corresponding phase distribution function is given in the entangled representation. In terms of this definition, we also analyze the phase behavior of some simple two-mode states such as squeezing-rotatlng coherent state, squeezing-rotating vacuum state, and so on. It is found that the results exactly agree with the foregoing phase theory.     

奇偶对相干态的维格纳函数和层析图函数

利用纠缠态η〉表象下的维格纳算符,重构了奇偶对相干态的维格纳函数.根据维格纳函数在相空间中随变量ρ和γ的变化规律,讨论了奇偶对相干态的非经典性质和量子干涉效应.研究发现,奇偶对相干态总呈现非经典性质,并且当q取奇数时,奇偶对相干态更容易出现非经典性质.奇偶对相干态的量子干涉效应的显著程度与q取值有关,但对于q的同一取值,奇对相干态的量子干涉效应更为显著.利用纠缠态η〉表象下的维格纳算符Δ1,2(ρ,γ)和纠缠态η,τ1,τ2〉的投影算符之间满足的拉东变换,获得了奇偶对相干态的量子层析图函数.     

Asymptotic solution for a perturbed mechanism of western boundary undercurrents in the Pacific

This paper consider a class of perturbed mechanism for the western boundary undercurrents in the Pacific. The model of generalized governing equations is studied. Using the perturbation method, it constructs the asymptotic solution of the model. And the accuracy of asymptotic solution is proved by the theory of differential inequalities. Thus the uniformly valid asymptotic expansions of the solution are obtained.     

Phase Properties of Two-Mode Squeezing-Rotating Entangled Representation

By virtue of the squeezing-rotating entangled representation, we mainly establish thc new two-mode phase operator and phase angle operator, which is a general form including the foregoing formalist in two-mode Fock space.In addition, the corresponding phase distribution function is given in the entangled representation. In terms of this definition, we also analyze the phase behavior of some simple two-mode states such as squeezing-rotating coherent state,squeezing-rotating vacuum state, and so on. It is found that the results exactly agree with the foregoing phase theory.     

Unitary Transformation in kq Representation

In this paper, we discuss the unitary transformation induced by Z6 rotations of noncommutative space on the states （k, q,s）, which plays a key role in construction for noncommutative solitons T^2/Z6 by GHS method. As a result, we prove a well-known ＂Gauss Sum＂ formula in the number theory through a concise way.