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Dec 2000

Volume 41, Issue 12, pp. 7889-8355

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The onset of superconductivity in semi-infinite strips

Y. Almog

J. Math. Phys. 41, 7889 (2000); http://dx.doi.org/10.1063/1.1319857 (17 pages) | Cited 3 times

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The existence of a superconducting branch bifurcating from the normal state is proved in semi-infinite strips. It is proved that the critical magnetic field at which bifurcation takes place, or the onset field, for a semi-infinite strip is greater than the onset field for an infinite strip with the same width. In addition we find the loci of the vortices far away from the corners and show convergence of the bifurcating modes in long rectangles to those in the semi-infinite strip with the same width. © 2000 American Institute of Physics.
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74.25.Op Mixed states, critical fields, and surface sheaths
05.45.-a Nonlinear dynamics and chaos
74.25.Uv Vortex phases (includes vortex lattices, vortex liquids, and vortex glasses)

Inverse atomic densities and inequalities among density functionals

J. C. Angulo, E. Romera, and J. S. Dehesa

J. Math. Phys. 41, 7906 (2000); http://dx.doi.org/10.1063/1.1320857 (12 pages) | Cited 6 times

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Rigorous relationships among physically relevant quantities of atomic systems (e.g., kinetic, exchange, and electron–nucleus attraction energies, information entropy) are obtained and numerically analyzed. They are based on the properties of inverse functions associated to the one-particle density of the system. Some of the new inequalities are of great accuracy and/or improve similar ones previously known, and their validity extends to other many-fermion systems and to arbitrary dimensionality. © 2000 American Institute of Physics.
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31.15.E- Density-functional theory
05.30.Fk Fermion systems and electron gas
02.60.-x Numerical approximation and analysis

Eigenvalues in spectral gaps of the two-dimensional Pauli operator

Alexander Besch

J. Math. Phys. 41, 7918 (2000); http://dx.doi.org/10.1063/1.1289826 (14 pages)

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We consider purely magnetic two-dimensional Pauli operators H with a spectral gap, perturbed by a magnetic field λBs = λdmaths, λ ≥ 0. Assuming that Bs and maths vanish at infinity, we ask whether eigenvalues will cross the gap as λ→∞. Furthermore, we give an example of a two-dimensional Pauli operator H with periodic magnetic field of zero flux which has at least one spectral gap. © 2000 American Institute of Physics.
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03.65.Fd Algebraic methods
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

Multipartite generalization of the Schmidt decomposition

H. A. Carteret, A. Higuchi, and A. Sudbery

J. Math. Phys. 41, 7932 (2000); http://dx.doi.org/10.1063/1.1319516 (8 pages) | Cited 59 times

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We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class. © 2000 American Institute of Physics.
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03.65.-w Quantum mechanics

Group theoretical quantum tomography

G. Cassinelli, G. M. D’Ariano, E. De Vito, and A. Levrero

J. Math. Phys. 41, 7940 (2000); http://dx.doi.org/10.1063/1.1323497 (12 pages) | Cited 14 times

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The paper is devoted to the mathematical foundation of quantum tomography using the theory of square-integrable representations of unimodular Lie groups. © 2000 American Institute of Physics.
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03.65.Fd Algebraic methods
02.20.Qs General properties, structure, and representation of Lie groups
42.50.-p Quantum optics

Phase-integral formulas for quantal matrix elements

Per Olof Fröman

J. Math. Phys. 41, 7952 (2000); http://dx.doi.org/10.1063/1.1314896 (12 pages) | Cited 2 times

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Simple phase-integral formulas, not involving wave functions, are derived for quantal matrix elements associated with bound states of a quantal particle in a smooth single-well potential. In these formulas one uses an arbitrary order of the phase-integral approximation generated from an unspecified base function. © 2000 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states

Confluent hypergeometric equations and related solvable potentials in quantum mechanics

J. Negro, L. M. Nieto, and O. Rosas-Ortiz

J. Math. Phys. 41, 7964 (2000); http://dx.doi.org/10.1063/1.1323501 (33 pages) | Cited 14 times

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The connection between the Schrödinger and confluent hypergeometric equations is discussed. It is shown that the factorization of the confluent hypergeometric equation gives a unifying powerful algebraic tool in order to study some quantum mechanical eigenvalue problems. That description includes the linear and N-dimensional harmonic oscillators, as well as the Coulomb and Morse potentials. © 2000 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods

Matroid theory and Chern–Simons

J. A. Nieto and M. C. Marín

J. Math. Phys. 41, 7997 (2000); http://dx.doi.org/10.1063/1.1319518 (9 pages) | Cited 7 times

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It is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang–Mills physics, but also for M-theory. Our discussion is focused in an action consisting purely of the Chern–Simons term, but in principle the main ideas can be applied beyond such an action. In our treatment the theorem due to Thistlethwaite, which gives a relationship between the Tutte polynomial for graphs and Jones polynomial for alternating knots and links, plays a central role. Before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the Fano matroid and D = 11 supergravity. Our approach also seems to be related to loop solutions of quantum gravity based in an Ashtekar formalism. © 2000 American Institute of Physics.
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11.15.-q Gauge field theories
04.65.+e Supergravity
12.40.Nn Regge theory, duality, absorptive/optical models
04.60.-m Quantum gravity
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
11.25.-w Strings and branes
11.30.-j Symmetry and conservation laws

On the absolutely continuous spectrum of Stark Hamiltonians

Jaouad Sahbani

J. Math. Phys. 41, 8006 (2000); http://dx.doi.org/10.1063/1.1287922 (10 pages) | Cited 3 times

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We study the spectral properties of the Schrödinger operator with a constant electric field perturbed by a bounded potential. It is shown that if the derivative of the potential in the direction of the electric field is smaller at infinity than the electric field, then the spectrum of the corresponding Stark operator is purely absolutely continuous. In one dimension, the absolute continuity of the spectrum is implied by just the boundedness of the derivative of the potential. The sharpness of our criterion for higher dimensions is illustrated by constructing smooth potentials with bounded partial derivatives for which the corresponding Stark operators have a dense point spectrum. © 2000 American Institute of Physics.
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78.20.Jq Electro-optical effects
71.70.Ej Spin-orbit coupling, Zeeman and Stark splitting, Jahn-Teller effect
03.65.Ge Solutions of wave equations: bound states

Ideal quantum gases in D-dimensional space and power-law potentials

Luca Salasnich

J. Math. Phys. 41, 8016 (2000); http://dx.doi.org/10.1063/1.1322078 (9 pages) | Cited 24 times

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We investigate ideal quantum gases in D-dimensional space and confined in a generic external potential by using the semiclassical approximation. In particular, we derive density of states, density profiles and critical temperatures for Fermions and Bosons trapped in isotropic power-law potentials. From such results, one can easily obtain those of quantum gases in a rigid box and in a harmonic trap. Finally, we show that the Bose–Einstein condensation can set up in a confining power-law potential if and only if D/2+D/n>1, where D is the space dimension and n is the power-law exponent. © 2000 American Institute of Physics.
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05.30.Jp Boson systems
05.30.Fk Fermion systems and electron gas

The semiclassical propagator for spin coherent states

Michael Stone, Kee-Su Park, and Anupam Garg

J. Math. Phys. 41, 8025 (2000); http://dx.doi.org/10.1063/1.1320856 (25 pages) | Cited 34 times

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We use a continuous-time path integral to obtain the semiclassical propagator for minimal-spread spin coherent states. We pay particular attention to the “extra phase” discovered by Solari and Kochetov, and show that this correction is related to an anomaly in the fluctuation determinant. We show that, once this extra factor is included, the semiclassical propagator has the correct short time behavior to O(T2), and demonstrate its consistency under dissection of the path. © 2000 American Institute of Physics.
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03.65.Sq Semiclassical theories and applications
03.65.Db Functional analytical methods
03.65.Fd Algebraic methods
02.30.Cj Measure and integration
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

Representation of quantum mechanical resonances in the Lax–Phillips Hilbert space

Y. Strauss, L. P. Horwitz, and E. Eisenberg

J. Math. Phys. 41, 8050 (2000); http://dx.doi.org/10.1063/1.1310359 (22 pages) | Cited 9 times

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We discuss the quantum Lax–Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax–Phillips S-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup associated with resonances, the decay law is exactly exponential. We explain how this profound difference between the quantum Lax–Phillips theory and the description of unstable systems in the framework of the standard quantum theory emerges. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax–Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. In the special case of a time-independent potential problem lifted trivially to the quantum Lax–Phillips theory, we show that the Lax–Phillips S-matrix is unitarily related to the S-matrix of standard scattering theory by a unitary transformation parametrized by the spectral variable σ of the Lax–Phillips theory. Analytic continuation in σ has some of the properties of a method developed some time ago for application to dilation analytic potentials. We work out an illustrative example of the theory using a Lee–Friedrichs model, which is generalized to a rank one potential in the Lax–Phillips Hilbert space. © 2000 American Institute of Physics.
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03.65.Nk Scattering theory
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
11.55.-m S-matrix theory; analytic structure of amplitudes

Generalized affine coherent states: A natural framework for the quantization of metric-like variables

Glenn Watson and John R. Klauder

J. Math. Phys. 41, 8072 (2000); http://dx.doi.org/10.1063/1.1286033 (11 pages) | Cited 5 times

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Affine variables, which have the virtue of preserving the positive-definite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations. © 2000 American Institute of Physics.
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04.60.Gw Covariant and sum-over-histories quantization
02.20.Sv Lie algebras of Lie groups
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A new superconformal mechanics

E. Deotto, G. Furlan, and E. Gozzi

J. Math. Phys. 41, 8083 (2000); http://dx.doi.org/10.1063/1.1323498 (25 pages) | Cited 3 times

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In this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector fields built over the symplectic space of the original system. Our supersymmetric Hamiltonian itself turns out to have a clear geometrical meaning being the Lie derivative of the Hamiltonian flow of conformal mechanics. Using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original nonsupersymmetric model. Besides the supersymmetric extension of the original Hamiltonian, we also provide the extension of the other conformal generators present in the original system. These extensions also have a supersymmetric character being the square of some Grassmannian charge. We build the whole superalgebra of these charges and analyze their closure. The representation of the even part of this superalgebra on the odd part turns out to be integer and not spinorial in character. © 2000 American Institute of Physics.
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45.20.Jj Lagrangian and Hamiltonian mechanics

The classical Kepler problem and geodesic motion on spaces of constant curvature

Aidan J. Keane, Richard K. Barrett, and John F. L. Simmons

J. Math. Phys. 41, 8108 (2000); http://dx.doi.org/10.1063/1.1324652 (9 pages) | Cited 3 times

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In this paper we clarify and generalize previous work by Moser and Belbruno concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as Hamiltonian systems and the phase flow in each system is characterized by the value of the corresponding Hamiltonian and one other parameter (the mass parameter in the Kepler problem and the curvature parameter in the geodesic motion problem). Using a canonical transformation the Hamiltonian vector field for the geodesic motion problem is transformed into one which is proportional to that for the Kepler problem. Within this framework the energy of the Kepler problem is equal to (minus) the curvature parameter of the constant curvature space and the mass parameter is given by the value of the Hamiltonian for the geodesic motion problem. We work with the corresponding family of evolution spaces and present a unified treatment which is valid for all values of energy continuously. As a result, there is a correspondence between the constants of motion for both systems and the Runge–Lenz vector in the Kepler problem arises in a natural way from the isometries of a space of constant curvature. In addition, the canonical nature of the transformation guarantees that the Poisson bracket Lie algebra of constants of motion for the classical Kepler problem is identical to that associated with geodesic motion on spaces of constant curvature. © 2000 American Institute of Physics.
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95.10.Ce Celestial mechanics (including n-body problems)
45.50.Pk Celestial mechanics
04.20.-q Classical general relativity
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Poisson algebras associated with constrained dispersionless modified Kadomtsev–Petviashvili hierarchies

Jen-Hsu Chang and Ming-Hsien Tu

J. Math. Phys. 41, 8117 (2000); http://dx.doi.org/10.1063/1.1322080 (15 pages) | Cited 4 times

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We investigate the bi-Hamiltonian structures associated with constrained dispersionless modified Kadomtsev–Petviashvili (KP) hierarchies which are constructed from truncations of the Lax operator of the dmKP hierarchy. After transforming their second Hamiltonian structures to those of the Gelfand–Dickey-type, we obtain the Poisson algebras of the coefficient functions of the truncated Lax operators. Then we study the conformal property and free-field realizations of these Poisson algebras. Some examples are worked out explicitly to illustrate the obtained results. © 2000 American Institute of Physics.
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02.10.-v Logic, set theory, and algebra
47.10.-g General theory in fluid dynamics

Integrability of the Cn and BCn Ruijsenaars–Schneider models

Kai Chen, Bo-yu Hou, and Wen-Li Yang

J. Math. Phys. 41, 8132 (2000); http://dx.doi.org/10.1063/1.1323502 (16 pages) | Cited 6 times

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We study the Cn and BCn Ruijsenaars–Schneider models with interaction potential of trigonometric and rational types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking a nonrelativistic limit, we also obtain the Lax pairs for the corresponding Calogero–Moser systems. © 2000 American Institute of Physics.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
11.15.-q Gauge field theories
02.30.Cj Measure and integration
05.30.-d Quantum statistical mechanics

Geometric phases for corotating elliptical vortex patches

B. N. Shashikanth and P. K. Newton

J. Math. Phys. 41, 8148 (2000); http://dx.doi.org/10.1063/1.1320855 (15 pages) | Cited 2 times

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We describe a geometric phase that arises when two elliptical vortex patches corotate. Using the Hamiltonian moment model of Melander, Zabusky, and Styczek [J. Fluid Mech. 167, 95–115 (1986)] we consider two corotating uniform elliptical patches evolving according to the second order truncated equations of the model. The phase is computed in the adiabatic setting of a slowly varying Hamiltonian as in the work of Hannay [J. Phys. A 18, 221–230 (1985)] and Berry [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We also discuss the geometry of the symplectic phase space of the model in the context of nonadiabatic phases. The adiabatic phase appears in the orientation angle of each patch—it is similiar in form and is calculated using a multiscale perturbation procedure as in the point vortex configuration of Newton [Physica D 79, 416–423 (1994)] and Shashikanth and Newton [J. Nonlinear Sci. 8, 183–214 (1998)], however, an extra factor due to the internal stucture of the patch is present. The final result depends on the initial orientation of the patches unlike the phases in the works of Hannay and Berry [J. Phys. A 18, 221–230 (1985)]; [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle π:T2×S1→S1, where T2 is identified with the product of the momentum level sets of two Kirchhoff vortex patches and S1 is diffeomorphic to the momentum level set of two point vortex motion. This two point vortex motion is the motion that the patch centroids approach in the adiabatic limit. © 2000 American Institute of Physics.
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47.32.C- Vortex dynamics
02.40.-k Geometry, differential geometry, and topology
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Multiply warped products with nonsmooth metrics

Jaedong Choi

J. Math. Phys. 41, 8163 (2000); http://dx.doi.org/10.1063/1.1287432 (7 pages) | Cited 5 times

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In this article we study manifolds with C0-metrics and properties of Lorentzian multiply warped products. We represent the interior Schwarzschild space–time as a multiply warped product space–time with warping functions and we also investigate the curvature of a multiply warped product with C0-warping functions. We give the Ricci curvature in terms of f1, f2 for the multiply warped products of the form M = (0,2m)×f1R1×f2S2. © 2000 American Institute of Physics.
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04.20.Jb Exact solutions
02.40.Ma Global differential geometry
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E6 unification model building II. Clebsch–Gordan coefficients of 78⊗78

Gregory W. Anderson and Tomáš Blažek

J. Math. Phys. 41, 8170 (2000); http://dx.doi.org/10.1063/1.1308077 (20 pages) | Cited 3 times

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We have computed the Clebsch–Gordan coefficients for the product (000 001)⊗(000 001), where (000 001) is the adjoint 78-dimensional representation of E6. The results are presented for the dominant weights of the irreducible representations in this product. As a simple application we express the singlet operator in 27⊗78⊗math in terms of multiplets of the Standard Model gauge group. © 2000 American Institute of Physics.
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12.60.-i Models beyond the standard model
11.30.Ly Other internal and higher symmetries
11.15.-q Gauge field theories
02.20.Sv Lie algebras of Lie groups

A homogeneous space–time model with singularities

Shirley Bromberg and Alberto Medina

J. Math. Phys. 41, 8190 (2000); http://dx.doi.org/10.1063/1.1320858 (6 pages) | Cited 2 times

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We show the existence of a left invariant pseudo-Riemannian metric on the Oscillator Group of dimension four, such that this group becomes a space–time with singularities in the sense of Hawking and Penrose. As an application we exhibit new incomplete, nonhomeomorphic compact Lorentzian manifolds. © 2000 American Institute of Physics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
02.40.Hw Classical differential geometry
02.40.Ma Global differential geometry

Algebraic solutions for all dihedral groups

Jin-Quan Chen, Peng-Dong Fan, Luke McAven, and Philip Butler

J. Math. Phys. 41, 8196 (2000); http://dx.doi.org/10.1063/1.1286513 (27 pages) | Cited 1 time

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The significant problem with using point groups is the dependence upon extensively tabulated results. We present simple algebraic expressions for the primitive characters, matrix irreps, symmetry adapted functions (SAFs), and Clebsch–Gordan coefficients for all dihedral groups, Dn, Cnv, Dnd, and Dnh. Those results, for arbitrary n and for single- and double-valued representations, have been derived in a simple manner without using group tables. Previously incomplete tabulated results are now redundant. In particular the parity dependence of the SAFs of the improper dihedral groups is shown analytically. Simple relations are derived between the SAFs of the proper and improper dihedral groups, so that the SAFs of the latter can be easily obtained from the SAFs of Dn. © 2000 American Institute of Physics.
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61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

A parametric limiting absorption problem with degeneration

D. Eidus

J. Math. Phys. 41, 8223 (2000); http://dx.doi.org/10.1063/1.1314893 (13 pages)

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We are concerned with acoustic waves propagation in inhomogeneous media in the case where density ρ(x), being independent of r = ∣x∣ and depending on a parameter τ, can vanish on some domains of math3 for some isolated value τ = τ0. We investigate the behavior of solutions of wave equation as τ→τ0, prove the limiting absorption principle for τ = τ0, and apply the obtained results to a wave fronts propagation problem. © 2000 American Institute of Physics.
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43.20.Bi Mathematical theory of wave propagation

Quasiperiodic and periodic solutions for vector nonlinear Schrödinger equations

J. C. Eilbeck, V. Z. Enolskii, and N. A. Kostov

J. Math. Phys. 41, 8236 (2000); http://dx.doi.org/10.1063/1.1318733 (13 pages) | Cited 8 times

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We consider quasiperiodic and periodic (cnoidal) wave solutions of a set of n-component vector nonlinear Schrödinger equations (VNLSEs). In a biased photorefractive crystal with a drift mechanism of nonlinear response and Kerr-type nonlinearity, n-component nonlinear Schrödinger equations can be used to model self-trapped mutually incoherent wave packets. These equations also model pulse–pulse interactions in wavelength-division-multiplexed channels of optical fiber transmission systems. Quasiperiodic wave solutions for the VNLSEs in terms of n-dimensional Kleinian functions are presented. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for n-component nonlinear Schrödinger equations are found. © 2000 American Institute of Physics.
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42.65.Jx Beam trapping, self-focusing and defocusing; self-phase modulation
02.10.De Algebraic structures and number theory
42.81.Dp Propagation, scattering, and losses; solitons

Symmetrized double quantum stochastic product integrals

R. L. Hudson and S. Pulmannová

J. Math. Phys. 41, 8249 (2000); http://dx.doi.org/10.1063/1.1323500 (14 pages) | Cited 2 times

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A theory is developed of product integrals of the form ∏a<s<b ∏c<t<d(1+g[h] (ds,dt)). Here [a,b[ and [c,d[ are disjoint finite subintervals of math+, and g[h] is a formal power series in the indeterminate h whose constant term is zero and whose coefficients are elements of L⊗L, where L is the space of basic differentials of a multidimensional quantum stochastic calculus. The product integrals are themselves formal power series in h whose coefficients are finite sums of iterated stochastic integrals against the elements of L. They are symmetrized in such a way that ∏a<s<b ∏c<t<d(1+g[h](ds,dt)) is the image, obtained by applying the representation J[a,b[⊗J[c,d[ to the coefficients, where J[a,b[ is the representation canonically associated with the interval [a,b[, of a formal power series ∏ ∏(1+dg[h]) whose coefficients lie in U⊗U, where U is the universal enveloping algebra of the Lie algebra L. It is shown that the naturally conjectured multiplication rule, analogous to the multiplication rule for simple product integrals, holds in the commutative case. © 2000 American Institute of Physics.
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03.65.Fd Algebraic methods
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
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