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May 2013

Volume 54, Issue 5 (partial)

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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

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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.
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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

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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.
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47.10.ad Navier-Stokes equations
02.60.-x Numerical approximation and analysis

Weak solutions for the Navier-Stokes equations with B∞∞−1(ln)+math+L2 initial data

Shangbin Cui

J. Math. Phys. 54, 051503 (2013); http://dx.doi.org/10.1063/1.4798790 (18 pages)

Online Publication Date: 21 May 2013

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One of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained. In the book “Recent Developments in the Navier-Stokes Problems,” Lemarié-Rieusset proved that the Navier-Stokes equations have global weak solutions for initial data in the space math(mathN)+L2(mathN) (0 < r < 1), where Xr is the space of functions whose pointwise products with Hr functions belong to L2, mathr denotes the closure of C0∞(mathN) in Xr, and math(mathN) is the Besov space over mathr. In this paper we partially extend this result of Lemarié-Rieusset to the larger initial value space B∞∞−1(ln)(mathN)+math(mathN)+L2(mathN) (0 < r < 1), where B∞∞−1(ln)(mathN) is a logarithmically modified version of the usual Besov space B∞∞−1(mathN).
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47.10.ad Navier-Stokes equations
02.30.Hq Ordinary differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems
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

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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.
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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

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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.
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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

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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.
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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

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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.
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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

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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.
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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

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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.
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03.65.Ge Solutions of wave equations: bound states
02.30.Uu Integral transforms

Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

M. V. Feigin, M. A. Hallnäs, and A. P. Veselov

J. Math. Phys. 54, 052106 (2013); http://dx.doi.org/10.1063/1.4804615 (22 pages)

Online Publication Date: 20 May 2013

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For the rational Baker-Akhiezer functions associated with special arrangements of hyperplanes with multiplicities we establish an integral identity, which may be viewed as a generalisation of the self-duality property of the usual Gaussian function with respect to the Fourier transformation. We show that the value of properly normalised Baker-Akhiezer function at the origin can be given by an integral of Macdonald-Mehta type and explicitly compute these integrals for all known Baker-Akhiezer arrangements. We use the Dotsenko-Fateev integrals to extend this calculation to all deformed root systems, related to the non-exceptional basic classical Lie superalgebras.
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02.20.Sv Lie algebras of Lie groups
02.30.Rz Integral equations
02.30.Uu Integral transforms
02.30.Nw Fourier analysis

On infinite-dimensional state spaces

Tobias Fritz

J. Math. Phys. 54, 052107 (2013); http://dx.doi.org/10.1063/1.4807079 (8 pages)

Online Publication Date: 22 May 2013

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It is well known that the canonical commutation relation [x, p] = i can be realized only on an infinite-dimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finite-dimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinite-dimensional model in which [x, p] = i holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinite-dimensionality. One drawback of using the canonical commutation relation for this purpose is that it has unclear operational meaning. Here, we identify an operationally well-defined context from which an analogous conclusion can be drawn: if two unitary transformations U, V on a quantum system satisfy the relation V−1U2V = U3, then finite-dimensionality entails the relation UV−1UV = V−1UVU; this implication strongly fails in some infinite-dimensional realizations. This is a result from combinatorial group theory for which we give a new proof. This proof adapts to the consideration of cases where the assumed relation V−1U2V = U3 holds only up to ε and then yields a lower bound on the dimension.
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03.65.Aa Quantum systems with finite Hilbert space
03.65.Fd Algebraic methods
02.10.Ox Combinatorics; graph theory
02.20.Uw Quantum groups
02.20.Tw Infinite-dimensional Lie groups
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

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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.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.67.-a Quantum information
03.65.Aa Quantum systems with finite Hilbert space

Quantum logarithmic Sobolev inequalities and rapid mixing

Michael J. Kastoryano and Kristan Temme

J. Math. Phys. 54, 052202 (2013); http://dx.doi.org/10.1063/1.4804995 (30 pages)

Online Publication Date: 21 May 2013

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A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative mathp-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed. This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants. Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained, including the depolarizing semigroup and quantum expanders.
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03.65.Db Functional analytical methods
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

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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.
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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

Supersymmetry algebra cohomology. IV. Primitive elements in all dimensions from D = 4 to D = 11

Friedemann Brandt

J. Math. Phys. 54, 052302 (2013); http://dx.doi.org/10.1063/1.4804953 (22 pages)

Online Publication Date: 20 May 2013

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The primitive elements of the supersymmetry algebra cohomology as defined in previous work are derived for standard supersymmetry algebras in dimensions D = 5, …, 11 for all signatures of the related Clifford algebras of gamma matrices and all numbers of supersymmetries. The results are presented in a uniform notation along with results of previous work for D = 4, and derived by means of dimensional extension from D = 4 up to D = 11.
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11.30.Pb Supersymmetry
02.10.Yn Matrix theory

The two-loop sunrise graph with arbitrary masses

Luise Adams, Christian Bogner, and Stefan Weinzierl

J. Math. Phys. 54, 052303 (2013); http://dx.doi.org/10.1063/1.4804996 (18 pages)

Online Publication Date: 22 May 2013

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We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve.
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02.10.Ox Combinatorics; graph theory
02.30.Rz Integral equations
02.30.Hq Ordinary differential equations
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

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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.
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04.20.Gz Spacetime topology, causal structure, spinor structure
95.30.Sf Relativity and gravitation
02.40.-k Geometry, differential geometry, and topology

Spinors and the Weyl tensor classification in six dimensions

Carlos Batista and Bruno Carneiro da Cunha

J. Math. Phys. 54, 052502 (2013); http://dx.doi.org/10.1063/1.4804991 (25 pages)

Online Publication Date: 21 May 2013

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A spinorial approach to six-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the four-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the Goldberg-Sachs theorem. We work in complexified spaces, showing that the results for any signature can be obtained by taking the desired real slice.
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11.30.Ly Other internal and higher symmetries
04.20.Jb Exact solutions
02.10.Ud Linear algebra
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

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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.
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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

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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.
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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

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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)+.
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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

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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.
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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.)

Cosmic-ray diffusion modeling: Solutions using variational methods

R. C. Tautz and I. Lerche

J. Math. Phys. 54, 053303 (2013); http://dx.doi.org/10.1063/1.4806649 (14 pages)

Online Publication Date: 22 May 2013

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The diffusion of energetic particles in turbulent magnetic fields is usually described via the two-point, two-time velocity correlation function. A variational principle is used to determine the characteristic function that results from the Fourier-transformed correlation function. Both for a linear approximation and for the wave vector set to zero, explicit solutions are derived that depend on the Fokker-Planck coefficient of pitch-angle scattering. It is shown that, for an isotropic form of the Fokker-Planck coefficient, the characteristic function is divergent, which can be remedied only by using a Fokker-Planck coefficient that is finite at all pitch angles.
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52.25.Fi Transport properties
52.35.Ra Plasma turbulence
02.30.Xx Calculus of variations
02.30.Uu Integral transforms
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

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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.
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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

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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.
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03.65.Nk Scattering theory
03.65.Db Functional analytical methods
03.65.Ge Solutions of wave equations: bound states
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