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Top 20 Most Read Articles
January 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Composite parameterization and Haar measure for all unitary and special unitary groups J. Math. Phys. 53, 013501 (2012); http://dx.doi.org/10.1063/1.3672064 (22 pages) Online Publication Date: 3 January 2012
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We adopt the concept of the composite parameterization of the unitary group U(d) to the special unitary group SU(d). Furthermore, we also consider the Haar measure in terms of the introduced parameters. We show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on U(d) and SU(d). With regard to possible applications of our results, we consider the computation of high-order integrals over unitary groups.
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The gap equation for spin-polarized fermions J. Math. Phys. 53, 012101 (2012); http://dx.doi.org/10.1063/1.3670747 (19 pages) Online Publication Date: 4 January 2012
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We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For cosh (δμ/T) ⩽ 2, with T the temperature and δμ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in [Frank, R. L., Hainzl, C., Naboko, S., and Seiringer, R., J., Geom. Anal. 17, 559–567 (2007)10.1007/BF02937429; Hainzl, C., Hamza, E., Seiringer, R., and Solovej, J. P., Commun., Math. Phys. 281, 349–367 (2008)10.1007/s00220-008-0489-2; and Hainzl, C. and Seiringer, R., Phys. Rev. B 77, 184517-110 435 (2008)]10.1103/PhysRevB.77.184517. For cosh (δμ/T) > 2 the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.
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Time‐Dependent Statistics of the Ising Model J. Math. Phys. 4, 294 (1963); http://dx.doi.org/10.1063/1.1703954 (14 pages) Online Publication Date: 22 December 2004
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The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N‐member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time‐varying magnetic field upon the model is discussed, and the frequency‐dependent magnetic susceptibility is found in the weak‐field limit. Some fluctuation‐dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time‐dependent correlation function of the magnetization at equilibrium. |
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Some Cluster Size and Percolation Problems J. Math. Phys. 2, 609 (1961); http://dx.doi.org/10.1063/1.1703745 (11 pages) Online Publication Date: 22 December 2004
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The problem of cluster size distribution and percolation on a regular lattice or graph of bonds and sites is reviewed and its applications to dilute ferromagnetism, polymer gelation, etc., briefly discussed. The cluster size and percolation problems are then solved exactly for Bethe lattices (infinite homogeneous Cayley trees) and for a wide class of pseudolattices derived by replacing the bonds and∕or sites of a Bethe lattice by arbitrary finite subgraphs. Explicit expressions are given for the critical probability (density), for the mean cluster size, and for the density of infinite clusters. The nature of the critical anomalies is shown to be the same for all lattices discussed; in particular, the density of infinite clusters vanishes as R(p) ≈ C(p−pc) (p≥pc). |
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Quasi-coherent states for harmonic oscillator with time-dependent parameters J. Math. Phys. 53, 012102 (2012); http://dx.doi.org/10.1063/1.3676072 (8 pages) Online Publication Date: 6 January 2012
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In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.
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Investigation on a nonisospectral fifth-order Korteweg-de Vries equation generalized from fluids J. Math. Phys. 53, 013502 (2012); http://dx.doi.org/10.1063/1.3673273 (8 pages) Online Publication Date: 4 January 2012
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In this paper, a nonisospectral fifth-order Korteweg-de Vries equation generalized from fluids is investigated. With symbolic computation, such equation is transformed into its bilinear form through a proposed dependent variable transformation with one more parameter than those in the existing literature. N-soliton solutions, Bäcklund transformation, and Lax pair in the explicit forms are constructed. Based on the above results, the characteristic-line method is applied to discuss the features of the solitons for the nonisospectral problem, i.e., the controllable solitonic velocities and widths. Four types of solitonic structures with the different solitonic velocities, widths, amplitudes, and backgrounds are also illustrated.
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J. Math. Phys. 6, 167 (1965); http://dx.doi.org/10.1063/1.1704269 (15 pages) Online Publication Date: 22 December 2004
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Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points. The number of distinct points visited after n steps on a k‐dimensional lattice (with k ≥ 3) when n is large is a1n + a2n☒ + a3 + a4n−☒ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face‐centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited. The probability F(c) that a walker on a one‐dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c∕(1 − c)] log c. Most of the results in this paper have been derived by the method of Green's functions. |
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An alternative approach to Schrödinger equations with a spatially varying mass J. Math. Phys. 52, 122103 (2011); http://dx.doi.org/10.1063/1.3672208 (5 pages) Online Publication Date: 23 December 2011
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Extending the point canonical transformation approach in a manner distinct from the previous ones, we propose a unified approach of generating potentials of all classes having non-constant masses. |
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Exponential Operators and Parameter Differentiation in Quantum Physics J. Math. Phys. 8, 962 (1967); http://dx.doi.org/10.1063/1.1705306 (21 pages) Online Publication Date: 21 December 2004
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Elementary parameter‐differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker‐Campbell‐Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)∕dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated‐commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed‐form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed‐form single‐exponential solutions to the differential equation dY(t)∕dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple‐exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic‐oscillator canonical‐ensemble averages of general operators defined by the Weyl prescription. Solutions of the density‐matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian. |
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Some models of anisotropic spheres in general relativity J. Math. Phys. 22, 118 (1981); http://dx.doi.org/10.1063/1.524742 (8 pages)
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A heuristic procedure is developed to obtain interior solutions of Einstein’s equations for anisotropic matter from known solutions for isotropic matter. Five known solutions are generalized to give solutions with anisotropic sources. |
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The stochastisation hypothesis and the spacing of planetary systems J. Math. Phys. 52, 113502 (2011); http://dx.doi.org/10.1063/1.3658279 (20 pages) Online Publication Date: 8 November 2011
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We introduce the stochastisation hypothesis which aims to provide a framework to deal with physical systems in random environment. We apply the stochastisation hypothesis in two different cases: in the study of the dynamics of a protoplanetary nebula and in the chaotic long-term behaviour of a generic planetary system using previous works of Albeverio et al. [“A stochastic model for the orbits of planets and satellites: an interpretation of Titius-Bode law,” Expo. Math. 4, 363–373 (1983)], Nottale [“The quantization of the solar system,” Astron. Astrophys. 315, L9 (1996)], and Laskar [“On the spacing of planetary systems,” Phys. Rev. Lett. 84(15), 3240–3243 (2000)10.1103/PhysRevLett.84.3240]. These results give both a particular law for the distribution of planetary orbits. |
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Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials J. Math. Phys. 52, 122107 (2011); http://dx.doi.org/10.1063/1.3671966 (8 pages) Online Publication Date: 30 December 2011
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We show how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum, and Darboux-Bäcklund transformations.
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J. Math. Phys. 50, 023510 (2009); http://dx.doi.org/10.1063/1.3077223 (10 pages) Online Publication Date: 13 February 2009
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In this paper, we studied the Boussinesq-like equations with fully nonlinear dispersion B(m,n) equations which exhibit solutions with solitary patterns. New exact solitary solutions of the equations are found. The two special cases, B(2,2) and B(3,3), are chosen to illustrate the concrete scheme of the homotopy perturbation method in B(m,n) equations. The nonlinear equations B(m,n) are addressed for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of B(m,n) equations are established.
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Super extension of Bell polynomials with applications to supersymmetric equations J. Math. Phys. 53, 013503 (2012); http://dx.doi.org/10.1063/1.3673275 (18 pages) Online Publication Date: 4 January 2012
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In this paper, we generalize classical Bell polynomials into super version, which are found to be effective in systematically constructing super bilinear representation, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws of supersymmetric equations. We take N = 1 supersymmetric KdV equation and N = 2 supersymmetric sine-Gordon equation to illustrate this procedure.
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On the supremum and infimum of bounded quantum observables J. Math. Phys. 52, 122101 (2011); http://dx.doi.org/10.1063/1.3671331 (6 pages) Online Publication Date: 20 December 2011
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Let S(H) be the set of all bounded self-adjoint linear operators on a complex Hilbert space H. In 2006, Gudder [Math. Slovaca 56, 573 (2006)] introduced a new order ≼ on S(H). Since then, the existence conditions and representations of the supremum and infimum of two elements in S(H) with respect to the order ≼ have been intensively studied. Specifically, Li and Sun [J. Math. Phys. 50, 122107 (2009)]10.1063/1.3272542 obtained simpler representations of A ∧ P and A ∨ P, where A∈S(H) and P is an orthogonal projection on H. In this note, we present more intuitive and concise results on A ∨ P and extend the results of Li and Sun to more general cases. Moreover, some applications of our results are given to show that our results are easier to deal with.
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The Einstein Tensor and Its Generalizations J. Math. Phys. 12, 498 (1971); http://dx.doi.org/10.1063/1.1665613 (4 pages) Online Publication Date: 28 October 2003
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The Einstein tensor Gij is symmetric, divergence free, and a concomitant of the metric tensor gab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors. |
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The propagator of the attractive delta-Bose gas in one dimension J. Math. Phys. 52, 122106 (2011); http://dx.doi.org/10.1063/1.3663431 (17 pages) Online Publication Date: 30 December 2011
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We consider the quantum δ-Bose gas on the infinite line. For repulsive interactions, Tracy and Widom have obtained an exact formula for the quantum propagator. In our contribution we explicitly perform its analytic continuation to attractive interactions. We also study the connection to the expansion of the propagator in terms of the Bethe ansatz eigenfunctions. Thereby we provide an independent proof of their completeness.
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J. Math. Phys. 51, 054101 (2010); http://dx.doi.org/10.1063/1.3414818 (3 pages) Online Publication Date: 6 May 2010
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We comment on the recent paper of
Di Criscienzo and Zerbini [J. Math. Phys. 50, 103517 (2009)]
. We argue that the Euclidean evolution operator computed in our paper (
K. Andrzejewski et al., e-print arXiv:0904.3055
) is correct contrary to the claim of Di Criscienzo and Zerbini.
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Statistical Theory of the Energy Levels of Complex Systems. I J. Math. Phys. 3, 140 (1962); http://dx.doi.org/10.1063/1.1703773 (17 pages) Online Publication Date: 22 December 2004
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New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time‐reversal invariance is violated, and the symplectic ensemble applies only to odd‐spin systems without rotational symmetry. The probability‐distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one‐dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group‐theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper. |
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J. Math. Phys. 53, 012901 (2012); http://dx.doi.org/10.1063/1.3676075 (12 pages) Online Publication Date: 6 January 2012
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We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painlevé transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.
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