JMP Nameplate
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

Flickr Twitter iResearch App Facebook

BROWSE VOLUMES

Year Range: 
Search Issue | RSS Feeds RSS
Previous Issue

May 2013

Volume 54, Issue 5 (partial)

back to top
RSS Feeds
back to top Partial Differential Equations

Exact wave solutions for Bose–Einstein condensates with time-dependent scattering length and spatiotemporal complicated potential

E. Kengne, A. Lakhssassi, R. Vaillancourt, and Wu-Ming Liu

J. Math. Phys. 54, 051501 (2013); http://dx.doi.org/10.1063/1.4803458 (14 pages)

Online Publication Date: 7 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We consider a cubic-quintic Gross–Pitaevskii equation which governs the dynamics of Bose–Einstein condensate matter waves with time-dependent scattering length and spatiotemporal complex potential. By introducing phase-imprint parameters in the system, we present the integrable condition for the equation and obtain the exact analytical solutions, which describe the propagation of a solitary wave. By applying specific time-modulated feeding/loss functional parameter, various types of magnetic trap strengths, and phase-imprint parameters, the dynamics of the solutions can be controlled. Solitary wave solutions with breathing and snaking behaviors are reported.
Show PACS
03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations
05.45.Yv Solitons

On the strong solutions of one-dimensional Navier-Stokes-Poisson equations for compressible non-Newtonian fluids

Yukun Song, Hongjun Yuan, and Yang Chen

J. Math. Phys. 54, 051502 (2013); http://dx.doi.org/10.1063/1.4803485 (16 pages)

Online Publication Date: 8 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We study the strong solutions of 1D Navier-Stokes-Poisson equations for compressible non-Newtonian fluids in bounded intervals. The model is raised from the viscous isentropic gas flow under considering an external force and the non-Newtonian gravitational force term. By using the iterative method we prove the local existence and uniqueness of strong solutions based on some compatibility condition. The main condition is that the initial density vacuum is allowed.
Show PACS
47.10.ad Navier-Stokes equations
02.60.-x Numerical approximation and analysis
back to top Representation Theory and Algebraic Methods

Real second order freeness and Haar orthogonal matrices

James A. Mingo and Mihai Popa

J. Math. Phys. 54, 051701 (2013); http://dx.doi.org/10.1063/1.4804168 (35 pages)

Online Publication Date: 14 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier.
Show PACS
02.10.Ud Linear algebra
02.60.Dc Numerical linear algebra
02.10.Yn Matrix theory
back to top Quantum Mechanics

Hartman effect and dissipative quantum systems

Samyadeb Bhattacharya and Sisir Roy

J. Math. Phys. 54, 052101 (2013); http://dx.doi.org/10.1063/1.4803132 (7 pages)

Online Publication Date: 2 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The dwell time for dissipative quantum system is shown to increase with barrier width. It clearly precludes Hartman effect for dissipative systems. Here calculation has been done for inverted parabolic potential barrier.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory
02.30.-f Function theory, analysis

Coulomb problem in non-commutative quantum mechanics

Veronika Gáliková and Peter Prešnajder

J. Math. Phys. 54, 052102 (2013); http://dx.doi.org/10.1063/1.4803457 (20 pages)

Online Publication Date: 6 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The aim of this paper is to find out how it would be possible for space non-commutativity (NC) to alter the quantum mechanics (QM) solution of the Coulomb problem. The NC parameter λ is to be regarded as a measure of the non-commutativity – setting λ = 0 which means a return to the standard quantum mechanics. As the very first step a rotationally invariant NC space Rλ3, an analog of the Coulomb problem configuration space (R3 with the origin excluded) is introduced. Rλ3 is generated by NC coordinates realized as operators acting in an auxiliary (Fock) space F. The properly weighted Hilbert-Schmidt operators in F form Hλ, a NC analog of the Hilbert space of the wave functions. We will refer to them as “wave functions”   also in the NC case. The definition of a NC analog of the hamiltonian as a hermitian operator in Hλ is one of the key parts of this paper. The resulting problem is exactly solvable. The full solution is provided, including formulas for the bound states for E < 0 and low-energy scattering for E > 0 (both containing NC corrections analytic in λ) and also formulas for high-energy scattering and unexpected bound states at ultra-high energy (both containing NC corrections singular in λ). All the NC contributions to the known QM solutions either vanish or disappear in the limit λ → 0.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Aa Quantum systems with finite Hilbert space
02.40.Gh Noncommutative geometry
03.65.Ge Solutions of wave equations: bound states

Spherical Schrödinger operators with δ-type interactions

Sergio Albeverio, Aleksey Kostenko, Mark Malamud, and Hagen Neidhardt

J. Math. Phys. 54, 052103 (2013); http://dx.doi.org/10.1063/1.4803708 (24 pages)

Online Publication Date: 13 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We investigate spectral properties of spherical Schrödinger operators (also known as Bessel operators) with δ-point interactions concentrated on a discrete set. We obtain necessary and sufficient conditions for these Hamiltonians to be self-adjoint, lower-semibounded and also we investigate their spectra. We also extend the classical Bargmann estimate to such Hamiltonians. In certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities. We apply our results to Schrödinger operators in L2(mathn) with a singular interaction supported by an infinite family of concentric spheres.
Show PACS
03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
03.65.Fd Algebraic methods

Thermal nonlinear coherent states on a flat space and on a sphere

H. Bagheri and A. Mahdifar

J. Math. Phys. 54, 052104 (2013); http://dx.doi.org/10.1063/1.4804357 (12 pages)

Online Publication Date: 15 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
In this paper, we first define thermal nonlinear coherent states on a sphere and show that these states are essentially two-mode squeezed nonlinear coherent states of the sphere at zero temperature. Then we consider quantum statistical properties of the thermal sphere nonlinear coherent states. In particular, we investigate temperature effects on transition of the constructed states from nonclassical states to classical ones. By using the Mandel parameter, we obtain a transition temperature and show that this transition temperature increases by increasing the curvature of the physical space. It turns out that, increasing curvature of the space provides nonlinear coherent states with nonclassical properties in higher temperature ranges.
Show PACS
05.30.-d Quantum statistical mechanics
05.70.Ce Thermodynamic functions and equations of state

Analytical solutions of the Schrödinger equation for a two-dimensional exciton in magnetic field of arbitrary strength

Ngoc-Tram Hoang-Do, Van-Hung Hoang, and Van-Hoang Le

J. Math. Phys. 54, 052105 (2013); http://dx.doi.org/10.1063/1.4804616 (11 pages)

Online Publication Date: 16 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The Feranchuk-Komarov operator method is developed by combining with the Levi-Civita transformation in order to construct analytical solutions of the Schrödinger equation for a two-dimensional exciton in a uniform magnetic field of arbitrary strength. As a result, analytical expressions for the energy of the ground and excited states are obtained with a very high precision of up to four decimal places. Especially, the precision is uniformly stable for the whole range of the magnetic field. This advantage appears due to the consideration of the asymptotic behaviour of the wave-functions in strong magnetic field. The results could be used for various physical analyses and the method used here could also be applied to other atomic systems.
Show PACS
03.65.Ge Solutions of wave equations: bound states
02.30.Uu Integral transforms
back to top Quantum Information and Computation

Approximately clean quantum probability measures

Douglas Farenick, Remus Floricel, and Sarah Plosker

J. Math. Phys. 54, 052201 (2013); http://dx.doi.org/10.1063/1.4803682 (15 pages)

Online Publication Date: 10 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
A quantum probability measure–or quantum measurement–is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al. [“Clean positive operator valued measures,” J. Math. Phys. 46(8), 082109 (2005)10.1063/1.2008996] for finite-dimensional Hilbert space, and was studied subsequently by Kahn [“Clean positive operator-valued measures for qubits and similar cases,” J. Phys. A 40(18), 4817–4832 (2007)10.1088/1751-8113/40/18/009] and Pellonpää [“Complete characterization of extreme quantum observables in finite dimensions,” J. Phys. A 44(8), 085304 (2011)10.1088/1751-8113/44/8/085304]. The present paper provides new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of “approximately clean quantum probability measures” and characterise this property for measures whose range determines a finite-dimensional operator system.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory
03.67.-a Quantum information
03.65.Aa Quantum systems with finite Hilbert space
back to top Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Bohr-Sommerfeld quantization condition for Dirac states derived from an Ermakov-type invariant

Karl-Erik Thylwe and Patrick McCabe

J. Math. Phys. 54, 052301 (2013); http://dx.doi.org/10.1063/1.4803030 (7 pages)

Online Publication Date: 2 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
It is shown that solutions of the second-order decoupled radial Dirac equations satisfy Ermakov-type invariants. These invariants lead to amplitude-phase-type representations of the radial spinor solutions, with exact relations between their amplitudes and phases. Implications leading to a Bohr-Sommerfeld quantization condition for bound states, and a few particular atomic/ionic and nuclear/hadronic bound-state situations are discussed.
Show PACS
03.65.Ge Solutions of wave equations: bound states
03.65.Pm Relativistic wave equations
03.65.Ta Foundations of quantum mechanics; measurement theory
02.10.Ud Linear algebra
back to top General Relativity and Gravitation

Lifting general relativity to observer space

Steffen Gielen and Derek K. Wise

J. Math. Phys. 54, 052501 (2013); http://dx.doi.org/10.1063/1.4802878 (29 pages)

Online Publication Date: 7 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The “observer space” of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein's equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.
Show PACS
04.20.Gz Spacetime topology, causal structure, spinor structure
95.30.Sf Relativity and gravitation
02.40.-k Geometry, differential geometry, and topology
back to top Dynamical Systems

A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation

Xiaoping Yuan and Kangkang Zhang

J. Math. Phys. 54, 052701 (2013); http://dx.doi.org/10.1063/1.4803852 (23 pages)

Online Publication Date: 15 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
In this paper, we consider the time dependent Schrödinger operator with a temporal quasi-periodic perturbation which is some function of class Cℓ for any ℓ ⩾ 100(3n + 2τ + 1), by using the KAM technique, we establish a reduction theorem for the above operator and prove that the invariant tori of the nearly-integrable Hamiltonian system with unbounded small perturbation are linearly stable both for math<d−1 and the limiting case math = d−1.
Show PACS
03.65.Ge Solutions of wave equations: bound states
02.70.Bf Finite-difference methods
02.30.Sa Functional analysis
back to top Classical Mechanics and Classical Fields

Mathematical structure of unit systems

Masao Kitano

J. Math. Phys. 54, 052901 (2013); http://dx.doi.org/10.1063/1.4802876 (17 pages)

Online Publication Date: 3 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We investigate the mathematical structure of unit systems and the relations between them. Looking over the entire set of unit systems, we can find a mathematical structure that is called preorder (or quasi-order). For some pair of unit systems, there exists a relation of preorder such that one unit system is transferable to the other unit system. The transfer (or conversion) is possible only when all of the quantities distinguishable in the latter system are always distinguishable in the former system. By utilizing this structure, we can systematically compare the representations in different unit systems. Especially, the equivalence class of unit systems (EUS) plays an important role because the representations of physical quantities and equations are of the same form in unit systems belonging to an EUS. The dimension of quantities is uniquely defined in each EUS. The EUS’s form a partially ordered set. Using these mathematical structures, unit systems and EUS’s are systematically classified and organized as a hierarchical tree.
Show PACS
03.50.De Classical electromagnetism, Maxwell equations
02.10.Ab Logic and set theory
02.10.Ox Combinatorics; graph theory
back to top Statistical Physics

A characterization of the Helmholtz free energy

Abolfazl Sanami

J. Math. Phys. 54, 053301 (2013); http://dx.doi.org/10.1063/1.4802749 (7 pages)

Online Publication Date: 1 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
Let H be a finite dimensional complex Hilbert space, math(H)+ be the set of all positive semi-definite operators (matrices) on H and φ is a (not necessarily linear) map of math(H)+ preserving the generalized Helmholtz free energy. In this paper, under suitable conditions we prove that there exists either a unitary or an anti-unitary operator U on H such that φ(A) = UAU* for any A∈math(H)+.
Show PACS
05.70.Ce Thermodynamic functions and equations of state
02.30.Tb Operator theory

A decomposition of irreversible diffusion processes without detailed balance

Hong Qian

J. Math. Phys. 54, 053302 (2013); http://dx.doi.org/10.1063/1.4803847 (10 pages)

Online Publication Date: 8 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
As a generalization of deterministic, nonlinear conservative dynamical systems, a notion of canonical conservative dynamics with respect to a positive, differentiable stationary density ρ(x) is introduced: math = j(x) in which ∇·(ρ(x)j(x)) = 0. Such systems have a conserved “generalized free energy function” F[u] = ∫u(x, t)ln (u(x, t)/ρ(x))dx in phase space with a density flow u(x, t) satisfying ∂ut = −∇·(ju). Any general stochastic diffusion process without detailed balance, in terms of its Fokker-Planck equation, can be decomposed into a reversible diffusion process with detailed balance and a canonical conservative dynamics. This decomposition can be rigorously established in a function space with inner product defined as ⟨ϕ, ψ⟩ = ∫ρ−1(x)ϕ(x)ψ(x)dx. Furthermore, a law for balancing F[u] can be obtained: The non-positive dF[u(x, t)]/dt = Ein(t) − ep(t) where the “source” Ein(t) ⩾ 0 and the “sink” ep(t) ⩾ 0 are known as house-keeping heat and entropy production, respectively. A reversible diffusion has Ein(t) = 0. For a linear (Ornstein-Uhlenbeck) diffusion process, our decomposition is equivalent to the previous approaches developed by Graham and Ao, as well as the theory of large deviations. In terms of two different formulations of time reversal for a same stochastic process, the meanings of dissipative and conservative stationary dynamics are discussed.
Show PACS
05.60.-k Transport processes
05.45.-a Nonlinear dynamics and chaos
05.70.Ce Thermodynamic functions and equations of state
02.50.Ey Stochastic processes
02.30.-f Function theory, analysis
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
back to top Methods of Mathematical Physics

Stability of n-Lie homomorphisms and Jordan n-Lie homomorphisms on n-Lie algebras

Seong Sik Kim, John Michael Rassias, Yeol Je Cho, and Soo Hawn Kim

J. Math. Phys. 54, 053501 (2013); http://dx.doi.org/10.1063/1.4803026 (7 pages)

Online Publication Date: 1 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The motivation of this paper is to apply the Hypers-Ulam stability problems of some kinds of functional equations to the classes of n-Lie homomorphisms and n-Lie algebras by using the structures of n-Lie homomorphisms and n-Lie algebras. In this paper, the generalized Hyers-Ulam-Rassias stability of n-Lie homomorphisms and Jordan n-Lie homomorphisms on n-Lie algebras associated to the generalized Cauchy-Jensen-Rassias additive functional equation are investigated using the fixed point methods.
Show PACS
02.20.Sv Lie algebras of Lie groups

Coulomb-distorted plane wave: Partial wave expansion and asymptotic forms

I. Hornyak and A. T. Kruppa

J. Math. Phys. 54, 053502 (2013); http://dx.doi.org/10.1063/1.4803027 (7 pages)

Online Publication Date: 1 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
Partial wave expansion of the Coulomb-distorted plane wave is determined and studied. Dominant and sub-dominant asymptotic expansion terms are given and leading order three-dimensional asymptotic form is derived. The generalized hypergeometric function 2F2(a, a; a + l + 1, a − l; z) is expressed with the help of confluent hypergeometric functions and the asymptotic expansion of 2F2(a, a; a + l + 1, a − l; z) is simplified.
Show PACS
03.65.Nk Scattering theory
03.65.Db Functional analytical methods
03.65.Ge Solutions of wave equations: bound states

Schrödinger-Virasoro Lie conformal algebra

Yucai Su and Lamei Yuan

J. Math. Phys. 54, 053503 (2013); http://dx.doi.org/10.1063/1.4803029 (16 pages)

Online Publication Date: 2 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We first construct two new nonsimple conformal algebras associated with the Schrödinger-Virasoro Lie algebra and the extended Schrödinger-Virasoro Lie algebra, respectively. Then we study conformal derivations and free nontrivial rank one conformal modules of these two conformal algebras. Also, we provide a computation of their universal central extensions.
Show PACS
02.20.Sv Lie algebras of Lie groups

Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

Dorin Bucur and Pedro Freitas

J. Math. Phys. 54, 053504 (2013); http://dx.doi.org/10.1063/1.4803140 (6 pages)

Online Publication Date: 3 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We consider the problem of minimizing the kth Dirichlet eigenvalue of planar domains with fixed perimeter and show that, as k goes to infinity, the optimal domain converges to the ball with the same perimeter. We also consider this problem within restricted classes of domains such as n-polygons and tiling domains, for which we show that the optimal asymptotic domain is that which maximises the area for fixed perimeter within the given family, i.e., the regular n-polygon and the regular hexagon, respectively. Physically, the above problems correspond to the determination of the shapes within a given class which will support the largest number of modes below a given frequency.
Show PACS
02.60.Lj Ordinary and partial differential equations; boundary value problems
02.10.-v Logic, set theory, and algebra

Effective complex permittivity tensor of a periodic array of cylinders

Yuri A. Godin

J. Math. Phys. 54, 053505 (2013); http://dx.doi.org/10.1063/1.4803490 (9 pages)

Online Publication Date: 6 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We determine the effective complex permittivity of a two-dimensional composite, consisting of an arbitrary doubly periodic array of identical circular cylinders in a homogeneous matrix, and whose dielectric properties are complex-valued. Efficient formulas are provided to determine the effective complex permittivity tensor which are in excellent agreement with numerical calculations. We also show that in contrast to the real-valued case, the real and imaginary parts of the effective complex-valued tensor can exhibit non-monotonic behavior as functions of volume fraction of cylinders, and can be either greater or less than that of the constituents.
Show PACS
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
77.22.Ch Permittivity (dielectric function)
02.10.Ud Linear algebra
02.10.Yn Matrix theory

Classification of Lie point symmetries for quadratic Liénard type equation math+f(x)math2+g(x) = 0

Ajey K. Tiwari, S. N. Pandey, M. Senthilvelan, and M. Lakshmanan

J. Math. Phys. 54, 053506 (2013); http://dx.doi.org/10.1063/1.4803455 (19 pages)

Online Publication Date: 7 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Limathnard type equation, math+f(x)math2+g(x) = 0, where f(x) and g(x) are arbitrary functions of x. The symmetry analysis gets divided into two cases, (i) the maximal (eight parameter) symmetry group and (ii) non-maximal (three, two, and one parameter) symmetry groups. We identify the most general form of the quadratic Limathnard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the non-maximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations.
Show PACS
02.20.Rt Discrete subgroups of Lie groups
02.30.Hq Ordinary differential equations
02.30.Sa Functional analysis
02.30.Uu Integral transforms

The multi-dimensional Hamiltonian structures in the Whitham method

A. Ya. Maltsev

J. Math. Phys. 54, 053507 (2013); http://dx.doi.org/10.1063/1.4803856 (50 pages)

Online Publication Date: 9 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We consider the averaging of local field-theoretic Poisson brackets in the multi-dimensional case. As a result, we construct a local Poisson bracket for the regular Whitham system in the multidimensional situation. The procedure is based on the procedure of averaging of local conservation laws and follows the Dubrovin–Novikov scheme of the bracket averaging suggested in one-dimensional case. However, the features of the phase space of modulated parameters in higher dimensions lead to a different natural class of the averaged brackets in comparison with the one-dimensional situation. Here we suggest a direct procedure of construction of the bracket for the Whitham system for d > 1 and discuss the conditions of applicability of the corresponding scheme. At the end, we discuss canonical forms of the averaged Poisson bracket in the multidimensional case.
Show PACS
02.30.Jr Partial differential equations

Lie algebras and Hamiltonian structures of multi-component Ablowitz–Kaup–Newell–Segur hierarchy

Xiao-ying Zhu and Da-jun Zhang

J. Math. Phys. 54, 053508 (2013); http://dx.doi.org/10.1063/1.4804068 (13 pages)

Online Publication Date: 13 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
Isospectral and non-isospectral hierarchies of multi-component Ablowitz–Kaup–Newell–Segur (AKNS) are obtained from a matrix spectral problem, then by means of the zero curvature representations of the isospectral flows {Km} and non-isospectral flows {σn}, we construct the symmetries and their algebraic structures for isospectral multi-component AKNS hierarchies, demonstrate the recursive operator L is a strong and hereditary symmetry for the isospectral hierarchy. We also derive that there are implectic operator θ and symplectic operator J such that L = θJ, and discuss the multi-Hamiltonian structures and the Liouville integrability of the isospectral hierarchies.
Show PACS
02.20.Sv Lie algebras of Lie groups
02.30.Rz Integral equations

The two dimensional motion of a particle in an inverse square potential: Classical and quantum aspects

R. P. Martínez-y-Romero, H. N. Núñez-Yépez, and A. L. Salas-Brito

J. Math. Phys. 54, 053509 (2013); http://dx.doi.org/10.1063/1.4804356 (7 pages)

Online Publication Date: 17 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The classical 2D dynamics of a particle moving under an inverse square potential, −k/r2, is analysed. We show that such problem is an example of a geometric system since its negative energy orbits are equivalent to free motion on a certain hypersurface. We then solve in momentum space, the corresponding unrenormalized quantum problem showing that there is no discrete energy spectrum and, particularly, no ground state.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory
02.40.-k Geometry, differential geometry, and topology

On the relationship between the classical Dicke-Jaynes-Cummings-Gaudin model and the nonlinear Schrödinger equation

Dianlou Du and Xue Geng

J. Math. Phys. 54, 053510 (2013); http://dx.doi.org/10.1063/1.4804943 (18 pages)

Online Publication Date: 17 May 2013

Full Text: Read Online (HTML) | Download PDF

Show Abstract
In this paper, the relationship between the classical Dicke-Jaynes-Cummings-Gaudin (DJCG) model and the nonlinear Schrödinger (NLS) equation is studied. It is shown that the classical DJCG model is equivalent to a stationary NLS equation. Moreover, the standard NLS equation can be solved by the classical DJCG model and a suitably chosen higher order flow. Further, it is also shown that classical DJCG model can be transformed into the classical Gaudin spin model in an external magnetic field through a deformation of Lax matrix. Finally, the separated variables are constructed on the common level sets of Casimir functions and the generalized action-angle coordinates are introduced via the Hamilton-Jacobi equation.
Show PACS
03.65.Ge Solutions of wave equations: bound states
02.30.Hq Ordinary differential equations
03.65.Fd Algebraic methods
03.65.Db Functional analytical methods
Close

Google Calendar

ADVERTISEMENT

Go to AIP Home   © AIP Publishing LLC
close