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Dec 2001

Volume 42, Issue 12, pp. 5499-5919

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Scalar quantum field coupled to boundaries and to a background magnetic field

A. A. Actor and I. Bender

J. Math. Phys. 42, 5499 (2001); http://dx.doi.org/10.1063/1.1413521 (23 pages)

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The stationary states of a charged scalar quantum field interacting with a background consisting of both boundaries and a static magnetic field are investigated. Following the development of some general theory, the example of a uniform magnetic field perpendicular to two parallel Dirichlet boundaries is worked through in detail. © 2001 American Institute of Physics.
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11.10.-z Field theory
12.20.Ds Specific calculations

On the continuum limit of fermionic topological charge in lattice gauge theory

David H. Adams

J. Math. Phys. 42, 5522 (2001); http://dx.doi.org/10.1063/1.1415087 (12 pages) | Cited 20 times

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It is proved that the fermionic topological charge of SU(N) lattice gauge fields on the four-torus, given in terms of a spectral flow of the Hermitian Wilson–Dirac operator or, equivalently, as the index of the overlap Dirac operator, reduces to the continuum topological charge in the classical continuum limit when the parameter m0 is in the physical region 0<m0<2. © 2001 American Institute of Physics.
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11.15.Ha Lattice gauge theory
11.10.Cd Axiomatic approach
11.30.Ly Other internal and higher symmetries
02.40.Re Algebraic topology

Some navigation rules for D-brane monodromy

Paul S. Aspinwall

J. Math. Phys. 42, 5534 (2001); http://dx.doi.org/10.1063/1.1409963 (19 pages) | Cited 16 times

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We explore some aspects of monodromies of D-branes in the Kähler moduli space of Calabi–Yau compactifications. Here a D-brane is viewed as an object of the derived category of coherent sheaves. We compute all the interesting monodromies in some nontrivial examples and link our work to recent results and conjectures concerning helices and mutations. We note some particular properties of the 0-brane. © 2001 American Institute of Physics.
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11.27.+d Extended classical solutions; cosmic strings, domain walls, texture

Multidimensional phase space and sunset diagrams

A. Bashir, R. Delbourgo, and M. L. Roberts

J. Math. Phys. 42, 5553 (2001); http://dx.doi.org/10.1063/1.1416887 (12 pages) | Cited 7 times

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We derive expressions for the phase space of a particle of momentum p decaying into N particles, that are valid for any number of dimensions. These are the imaginary parts of so-called “sunset” diagrams, which we also obtain. The results are given as a series of hypergeometric functions, which terminate for odd dimensions and are also well suited for deriving the threshold behavior. © 2001 American Institute of Physics.
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11.25.-w Strings and branes
02.30.Lt Sequences, series, and summability
02.40.-k Geometry, differential geometry, and topology

Generalized Weyl–Wigner map and Vey quantum mechanics

Nuno Costa Dias and João Nuno Prata

J. Math. Phys. 42, 5565 (2001); http://dx.doi.org/10.1063/1.1415086 (15 pages) | Cited 11 times

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The Weyl–Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations, is still missing. In this article we present a covariant generalization of the Weyl order prescription and of the Weyl–Wigner map and use them to derive Vey quantum mechanics directly from the standard operator formulation. The procedure displays some interesting features: it yields all the key ingredients and provides a more straightforward interpretation of the Vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the Vey idiom. These features are illustrated through a simple example. © 2001 American Institute of Physics.
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03.65.Db Functional analytical methods

Spherically symmetric solutions of the sixth order SU(N) Skyrme models

I. Floratos and B. M. A. G. Piette

J. Math. Phys. 42, 5580 (2001); http://dx.doi.org/10.1063/1.1415742 (16 pages) | Cited 5 times

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Following the construction described by Ioannidou et al. [J. Math. Phys. 40, 6353 (1999)], we use the rational map ansatz to construct analytically some topologically nontrivial solutions of the generalized SU(3) Skyrme model defined by adding a sixth order term to the usual Lagrangian. These solutions are radially symmetric and some of them can be interpreted as bound states of Skyrmions. The same ansatz is used to construct low-energy configuration of the SU(N) Skyrme model. © 2001 American Institute of Physics.
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11.10.Lm Nonlinear or nonlocal theories and models
11.10.St Bound and unstable states; Bethe-Salpeter equations
11.30.Ly Other internal and higher symmetries
11.10.Ef Lagrangian and Hamiltonian approach

Gradient corrections for semiclassical theories of atoms in strong magnetic fields

Christian Hainzl

J. Math. Phys. 42, 5596 (2001); http://dx.doi.org/10.1063/1.1415744 (30 pages) | Cited 1 time

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This paper is divided into two parts. In the first one the von Weizsäcker term is introduced to the magnetic Thomas–Fermi theory and the resulting MTFW functional is mathematically analyzed. In particular, it is shown that the von Weizsäcker term produces the Scott correction up to magnetic fields of order B≪Z2, in accordance with a result of Ivrii on the quantum mechanical ground state energy. The second part is dedicated to gradient corrections for semiclassical theories of atoms restricted to electrons in the lowest Landau band. We consider modifications of the Thomas–Fermi theory for strong magnetic fields (STF), i.e., for B≪Z3. The main modification consists in replacing the integration over the variables perpendicular to the field by an expansion in angular momentum eigenfunctions in the lowest Landau band. This leads to a functional (DSTF) depending on a sequence of one-dimensional densities. For a one-dimensional Fermi gas the analogue of a Weizsäcker correction has a negative sign and we discuss the corresponding modification of the DSTF functional.© 2001 American Institute of Physics.
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31.15.bt Statistical model calculations (including Thomas-Fermi and Thomas-Fermi-Dirac models)
03.65.Sq Semiclassical theories and applications
05.30.Fk Fermion systems and electron gas

Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials

Thomas Hupfer, Hajo Leschke, and Simone Warzel

J. Math. Phys. 42, 5626 (2001); http://dx.doi.org/10.1063/1.1401138 (16 pages) | Cited 1 time

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We study a nonrelativistic charged particle on the Euclidean plane math2 subject to a perpendicular constant magnetic field and an math2-homogeneous random potential in the approximation that the corresponding random Landau Hamiltonian on the Hilbert space L2(math2) is restricted to the eigenspace of a single but arbitrary Landau level. For a wide class of math2-homogeneous Gaussian random potentials we rigorously prove that the associated restricted integrated density of states is absolutely continuous with respect to the Lebesgue measure. We construct explicit upper bounds on the resulting derivative, the restricted density of states. As a consequence, any given energy is seen to be almost surely not an eigenvalue of the restricted random Landau Hamiltonian. © 2001 American Institute of Physics.
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71.70.Di Landau levels
03.65.Ge Solutions of wave equations: bound states
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.-r Probability theory, stochastic processes, and statistics

Investigation of the relativistic equivalent Hamiltonian in the LS coupling scheme

R. Karazija and V. Jonauskas

J. Math. Phys. 42, 5642 (2001); http://dx.doi.org/10.1063/1.1415429 (10 pages) | Cited 5 times

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Armstrong’s method of relativistic description of atoms in LS coupling using the equivalent operator is developed. Its form in terms of standard unit operators acting within a space of one shell is obtained. The interpretation of separate terms of equivalent Breit operator is investigated in a general case of nonequivalent electrons. The relativistic Dirac–Fock equations for the level and term in LS coupling are derived. The equivalent operator is applied to obtain the averages of relativistic operators too. © 2001 American Institute of Physics.
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31.15.-p Calculations and mathematical techniques in atomic and molecular physics
31.30.J- Relativistic and quantum electrodynamic (QED) effects in atoms, molecules, and ions

Some exact results for mid-band and zero band-gap states of associated Lamé potentials

Avinash Khare and Uday Sukhatme

J. Math. Phys. 42, 5652 (2001); http://dx.doi.org/10.1063/1.1416487 (13 pages) | Cited 22 times

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Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lamé potentials, a(a+1)msn2(x,m)+b(b+1)m cn2(x,m)/dn2(x,m), consists of a finite number of bound bands followed by a continuum band when both a and b take integer values. Further, if a and b are unequal integers, we show that there must exist some zero band-gap states, i.e., doubly degenerate states with the same number of nodes. More generally, in case a and b are not integers, but either a+b or a−b is an integer (a ≠ b), we again show that several of the band-gaps vanish due to degeneracy of states with the same number of nodes. Finally, when either a or b is an integer and the other takes a half-integral value, we obtain exact analytic solutions for several mid-band states.© 2001 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states

Hofstadter butterfly as quantum phase diagram

D. Osadchy and J. E. Avron

J. Math. Phys. 42, 5665 (2001); http://dx.doi.org/10.1063/1.1412464 (7 pages) | Cited 9 times

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The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labeled by their (integer) Hall conductance, and a fractal structure. We describe various properties of this phase diagram: We establish Gibbs phase rules; count the number of components of each phase, and characterize the set of multiple phase coexistence.© 2001 American Institute of Physics.
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03.65.-w Quantum mechanics
71.10.Hf Non-Fermi-liquid ground states, electron phase diagrams and phase transitions in model systems
73.43.Nq Quantum phase transitions
05.45.Df Fractals
71.15.Ap Basis sets (LCAO, plane-wave, APW, etc.) and related methodology (scattering methods, ASA, linearized methods, etc.)

Coherent state path integral for the Bloch particle

Junya Shibata and Komajiro Niizeki

J. Math. Phys. 42, 5672 (2001); http://dx.doi.org/10.1063/1.1416489 (15 pages)

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We construct a coherent state path integral formalism for the one-dimensional Bloch particle within the single band model. The transition amplitude between two coherent states is a sum of transition amplitudes with different winding numbers on the two-dimensional phase space which has the same topology as that of the cylinder. Appearance of the winding number is due to the periodicity of the quasimomentum of the Bloch particle. Our formalism is successfully applied to a semiclassical motion of the Bloch particle under a uniform electric field. The wave packet exhibits not only the Bloch oscillation but also a similar breathing to the one for the squeezed state of a harmonic oscillator. © 2001 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
03.65.Sq Semiclassical theories and applications
03.65.Db Functional analytical methods
02.30.Cj Measure and integration

Möbius structure of the spectral space of Schrödinger operators with point interaction

Izumi Tsutsui, Tamás Fülöp, and Taksu Cheon

J. Math. Phys. 42, 5687 (2001); http://dx.doi.org/10.1063/1.1415432 (11 pages) | Cited 20 times

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The Schrödinger operator with point interaction in one dimension has a U(2) family of self-adjoint extensions. We study the spectrum of the operator and show that (i) the spectrum is uniquely determined by the eigenvalues of the matrix U ∊ U(2) that characterizes the extension, and that (ii) the space of distinct spectra is given by the orbifold T2/math2 which is a Möbius strip with boundary. We employ a parametrization of U(2) that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found. This allows us to find that (iii) physically distinct point interactions form a three-parameter quotient space of the U(2) family.© 2001 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra

Asymptotics for the condensate multivortex solutions in the self-dual Chern–Simons CP(1) model

Hee-Seok Nam

J. Math. Phys. 42, 5698 (2001); http://dx.doi.org/10.1063/1.1409962 (15 pages)

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In this paper we study the asymptotics for the condensate multivortex solutions in the self-dual Chern–Simons CP(1) model. When the breaking parameter s belongs to (−½,½), we show that for any sequence of multivortex solutions which lies between suitable super- and subsolutions with respect to the Chern–Simons coupling constant κ, we can find a subsequence which converges to a constant depending only on s as κ goes to zero. Also we investigate the locally uniform convergence speed. © 2001 American Institute of Physics.
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11.15.-q Gauge field theories
11.10.Lm Nonlinear or nonlocal theories and models
11.10.Cd Axiomatic approach
11.10.Jj Asymptotic problems and properties
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Partition asymptotics from one-dimensional quantum entropy and energy currents

Miles P. Blencowe and Nicholas C. Koshnick

J. Math. Phys. 42, 5713 (2001); http://dx.doi.org/10.1063/1.1416195 (5 pages) | Cited 6 times

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We give an alternative method to that of Hardy–Ramanujan–Rademacher to derive the leading exponential term in the asymptotic approximation to the partition function p(n,a), defined as the number of decompositions of a positive integer n into integer summands, with each summand appearing at most a times in a given decomposition. The derivation involves mapping to an equivalent physical problem concerning the quantum entropy and energy currents of particles flowing in a one-dimensional (1D) channel connecting thermal reservoirs, and which obey Gentile’s intermediate statistics with statistical parameter a. The method is also applied to partitions associated with Haldane’s fractional exclusion statistics. © 2001 American Institute of Physics.
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05.30.Ch Quantum ensemble theory
05.70.Ce Thermodynamic functions and equations of state

“Single ring theorem” and the disk-annulus phase transition

Joshua Feinberg, R. Scalettar, and A. Zee

J. Math. Phys. 42, 5718 (2001); http://dx.doi.org/10.1063/1.1412599 (23 pages) | Cited 8 times

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Recently, an analytic method was developed to study in the large N limit non-Hermitian random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the existing Gaussian non-Hermitian literature. One obtains an explicit algebraic equation for the integrated density of eigenvalues from which the Green’s function and averaged density of eigenvalues could be calculated in a simple manner. Thus, that formalism may be thought of as the non-Hermitian analog of the method due to Brézin, Itzykson, Parisi, and Zuber for analyzing Hermitian non-Gaussian random matrices. A somewhat surprising result is the so called “single ring” theorem, namely, that the domain of the eigenvalue distribution in the complex plane is either a disk or an annulus. In this article we extend previous results and provide simple new explicit expressions for the radii of the eigenvalue distribution and for the value of the eigenvalue density at the edges of the eigenvalue distribution of the non-Hermitian matrix in terms of moments of the eigenvalue distribution of the associated Hermitian matrix. We then present several numerical verifications of the previously obtained analytic results for the quartic ensemble and its phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we demonstrate numerically the “single ring” theorem for the sextic potential, namely, the potential of lowest degree for which the “single ring” theorem has nontrivial consequences. © 2001 American Institute of Physics.
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05.70.Fh Phase transitions: general studies
02.10.Yn Matrix theory
02.10.Ud Linear algebra
02.50.Cw Probability theory
05.40.Fb Random walks and Levy flights
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Dynamical systems embedded into Lie algebras

O. R. Campoamor-Stursberg, F. G. Gascon, and D. Peralta-Salas

J. Math. Phys. 42, 5741 (2001); http://dx.doi.org/10.1063/1.1412598 (12 pages)

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Analytical and geometrical information on certain dynamical systems X is obtained under the assumption that X is embedded into a certain real Lie algebra. © 2001 American Institute of Physics.
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45.20.-d Formalisms in classical mechanics
02.10.Ud Linear algebra

Nonintegrable reductions of the self-dual Yang–Mills equations in a metric of plane wave type

Devendra A. Kapadia

J. Math. Phys. 42, 5753 (2001); http://dx.doi.org/10.1063/1.1412466 (9 pages)

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Symmetry reductions of the self-dual Yang–Mills equations for SL(2,C) bundles with the background metric ds2 = 2 du dv−dx2+f2(u)dy2 are considered. One of the field components in the reduced equations can be cast into Jordan normal form after gauge transformations. The reduced equations for the two possible normal forms are equivalent, respectively, to certain generalizations of the Korteweg–de Vries (KdV) equation and the nonlinear Schrödinger (NLS) equation. It is shown that the generalized KdV and NLS equations fail the Painlevé test except when the metric is flat. The generalized KdV equation is transformed to a simple form in the case when f(u) = ua and it is shown that one may obtain either the KdV equation or the cylindrical KdV equation by this method only when the metric is flat. © 2001 American Institute of Physics.
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11.15.-q Gauge field theories

A family of integrable nonlinear equations of hyperbolic type

A. Tongas, D. Tsoubelis, and P. Xenitidis

J. Math. Phys. 42, 5762 (2001); http://dx.doi.org/10.1063/1.1416488 (23 pages) | Cited 9 times

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A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional reduction of the anti-self-dual Yang–Mills equations, is presented. It represents a generalization of the Ernst–Weyl equation of General Relativity related to colliding neutrino and gravitational waves, as well as of the fourth order equation of Schwarzian type related to the KdV hierarchies, which was introduced by Nijhoff, Hone, and Joshi recently. An auto-Bäcklund transformation of the new system is constructed, leading to a superposition principle remarkably similar to the one connecting four solutions of the KdV equation. At the level of the Ernst–Weyl equation, this Bäcklund transformation and the associated superposition principle yield directly a generalization of the single and double Harrison transformations of the Ernst equation, respectively. The very method of construction also allows for revealing, in an essentially algorithmic fashion, other integrability features of the main subsystems, such as their reduction to the Painlevé transcendents. © 2001 American Institute of Physics.
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11.15.-q Gauge field theories
02.30.-f Function theory, analysis
04.20.Jb Exact solutions
04.30.-w Gravitational waves
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Kaluza’s theory in generalized coordinates

Ana Laura García-Perciante, Alfredo Sandoval-Villalbazo, and L. S. García-Colín

J. Math. Phys. 42, 5785 (2001); http://dx.doi.org/10.1063/1.1412463 (15 pages)

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Maxwell’s equations can be obtained in generalized coordinates by considering the electromagnetic field as an external agent. The work presented here shows how to obtain the electrodynamics for a charged particle in generalized coordinates eliminating the concept of external force. Based on Kaluza’s formalism, the one presented here extends the 5×5 metric into a 6×6 space–time giving enough room to include magnetic monopoles in a very natural way. © 2001 American Institute of Physics.
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04.50.-h Higher-dimensional gravity and other theories of gravity
03.50.De Classical electromagnetism, Maxwell equations
14.80.Hv Magnetic monopoles

Electromagnetic propagators in hyperbolic Robertson–Walker cosmologies

Roman Tomaschitz

J. Math. Phys. 42, 5800 (2001); http://dx.doi.org/10.1063/1.1413522 (32 pages) | Cited 1 time

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Green functions (retarded, advanced, Feynman and Dyson propagators) are calculated for the electromagnetic field in Robertson–Walker cosmologies with hyperbolic 3-manifolds as spacelike slices. The starting point is the Proca equation, i.e., the Maxwell field with a finite photon mass for infrared regularization, in a static cosmology with simply connected hyperbolic 3-sections. The time and space components of the resolvent kernel are scalar and vectorial point-pair invariants, respectively, and this symmetry allows for an explicit evaluation in the spectral representation. It is found that the quantum propagators have a logarithmic infrared singularity, which drops out in the zero curvature limit. Retarded and advanced Green functions remain well defined in the limit of zero photon mass, and they admit a simple generalization, by conformal scaling, to expanding 3-spaces. In cosmologies with multiply connected hyperbolic 3-manifolds as spacelike sections, the four enumerated propagators are constructed by means of Poincaré series. The spectral decomposition of the Green functions is given in terms of Eisenstein series for a certain class of open hyperbolic 3-spaces, including those with Schottky covering groups corresponding to solid handle-bodies as spacelike slices. © 2001 American Institute of Physics.
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98.80.Jk Mathematical and relativistic aspects of cosmology
98.80.Qc Quantum cosmology
03.50.De Classical electromagnetism, Maxwell equations
02.30.-f Function theory, analysis
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Combinatorial identities for binary necklaces from exact ray-splitting trace formulas

R. Blümel and Yu. Dabaghian

J. Math. Phys. 42, 5832 (2001); http://dx.doi.org/10.1063/1.1413226 (8 pages) | Cited 5 times

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Based on an exact trace formula for a one-dimensional ray-splitting system, we derive novel combinatorial identities for cyclic binary sequences (Pólya necklaces). © 2001 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
02.10.-v Logic, set theory, and algebra

Democratic supersymmetry

Chandrashekar Devchand and Jean Nuyts

J. Math. Phys. 42, 5840 (2001); http://dx.doi.org/10.1063/1.1413523 (19 pages)

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We present generalizations of N-extended supersymmetry algebras in four dimensions, using Lorentz covariance and invariance under permutation of the N supercharges as selection criteria. © 2001 American Institute of Physics.
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11.30.Pb Supersymmetry
02.20.Sv Lie algebras of Lie groups
02.10.Ud Linear algebra

Sums of spherical waves for lattices, layers, and lines

S. Enoch, R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and J. N. Nixon

J. Math. Phys. 42, 5859 (2001); http://dx.doi.org/10.1063/1.1409348 (12 pages) | Cited 6 times

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We consider the connections between sums of spherical wave functions over lattices, layers, and lines. The differences between sums over lattices and those over a doubly periodic constituent layer are expressed in terms of series with exponential convergence. Correspondingly, sums over the layer can be regarded as composed of a sum over a central line, and another sum over displaced lines exhibiting exponential convergence. We exhibit formulas which can be used to calculate accurately and efficiently sums of spherical waves over lattices, layers, and lines, which in turn may be used to construct quasiperiodic Green’s functions for the Helmholtz equation, of use in scattering problems for layers and lines of spheres, and for finding the Bloch modes of lattices of spheres. We illustrate the numerical accuracy of our expressions. © 2001 American Institute of Physics.
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02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems

A note on the generalized fractal dimensions of a probability measure

Charles-Antoine Guérin

J. Math. Phys. 42, 5871 (2001); http://dx.doi.org/10.1063/1.1416194 (5 pages)

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We prove the following result on the generalized fractal dimensions Dq± of a probability measure μ on mathn. Let g be a complex-valued measurable function on mathn satisfying the following conditions: (1) g is rapidly decreasing at infinity, (2) g is continuous and nonvanishing at (at least) one point, (3) ∫ g ≠ 0. Define the partition function Λa(μ,q) = an(q−1)‖ga ∗ μmath, where ga(x) = a−ng(a−1x) and  ∗  is the convolution in mathn. Then for all q>1 we have Dq± = 1/(q−1)limr→0 infsup[log Λaμ(r,q)/log r]. © 2001 American Institute of Physics.
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02.50.Cw Probability theory
05.45.Df Fractals
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