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Dec 1996

Volume 37, Issue 12, pp. 5897-6590

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Factorization of scattering matrices due to partitioning of potentials in one‐dimensional Schrödinger‐type equations

Tuncay Aktosun, Martin Klaus, and Cornelis van der Mee

J. Math. Phys. 37, 5897 (1996); http://dx.doi.org/10.1063/1.531754 (19 pages) | Cited 9 times

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The one‐dimensional Schrödinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero‐energy transmission coefficient vanishes when the potential is nontrivial, but in the exceptional case this coefficient is nonzero, resulting in tunneling through the potential. It is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces. Furthermore, any nontrivial potential, generic or exceptional, can be fragmented into generic pieces in infinitely many ways. The results remain valid when Dirac delta functions are included in the potential and other coefficients are added to the Schrödinger equation. For such Schrödinger equations, factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem. © 1996 American Institute of Physics.
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02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
11.55.-m S-matrix theory; analytic structure of amplitudes
03.65.Nk Scattering theory
03.65.Ge Solutions of wave equations: bound states

lP interpolation and optimized bounds on pairwise interacting fermion systems

Jean‐Louis Basdevant and André Martin

J. Math. Phys. 37, 5916 (1996); http://dx.doi.org/10.1063/1.531756 (12 pages) | Cited 1 time

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We derive a set of inequalities which relate the translation invariant problem of N identical particles with pairwise interactions to an independent particle problem. These inequalities apply to attractive power law potentials V(r)=rq, 1≤q≤∞, and superpositions of such potentials; they become identities in the harmonic oscillator case q=2. We use the inequalities to derive new upper and lower bounds for the ground state energies of fermion systems, which interact through these potentials. These bounds improve all previous results in the range 1≤q≤∞; they reduce to the exact answer in the harmonic oscillator case. © 1996 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
05.30.Fk Fermion systems and electron gas

Interchannel resonances at a threshold

B. Baumgartner

J. Math. Phys. 37, 5928 (1996); http://dx.doi.org/10.1063/1.531757 (11 pages) | Cited 3 times

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Fermi’s Golden Rule, the perturbation theoretic formula for calculating the half‐width of a resonance, is not applicable to the case of a transition from a bound state into an open channel, when the energy of the bound state is exactly at the threshold of the continuum. We study solvable models of this phenomenon. The exact results coincide in leading order with the formulas found by modifying Fermi’s Golden Rule. © 1996 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states

Transmission of conduction electrons through a symmetric pair of delta‐barriers or delta‐wells embedded in a semiconductor or a metal

V. Bezák

J. Math. Phys. 37, 5939 (1996); http://dx.doi.org/10.1063/1.531758 (19 pages) | Cited 4 times

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The transmission coefficient T(k0) is calculated for conduction electrons incident with a wave vector k0 upon a double barrier (double well) formed of two equal delta‐barriers (of two equal delta‐wells) embedded in a one‐dimensional (1‐D) semiconductor or in a 1‐D metal. The stationary Schrödinger–Wannier equation E(−i∂/∂x)ψ+V(x)ψ=Eψ is solved for V(x)=γ[δ(x+a/2)+δ(x−a/2)] (with real and time‐independent parameters γ, a) and E=E(k0)>0. (The interband transitions are neglected.) The operator E(−i∂/∂x) corresponds to a given (possibly nonquadratic) dispersion function E(k) of the conduction electrons [E(0)=0]. It is shown that T(k0) is an oscillating function reaching the maximum value [T(k0)→1] on an infinite set {K(j)} of values of k0. The shape of T(k0) depends on the shape of the dispersion function E(k) in a simple way: T(k0)=Tpar(mv(k0)/ℏ)) where Tpar(k0) means the transmission coefficient in the special case when the dispersion function is quadratic, Epar(k)=ℏ2k2/2m, and v(k)=(1/ℏ)∂E(k)/∂k is the group velocity due to E(k). [Here E(k) is taken as an increasing function.] © 1996 American Institute of Physics.
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72.15.Nj Collective modes (e.g., in one-dimensional conductors)
72.20.Fr Low-field transport and mobility; piezoresistance
02.10.Ab Logic and set theory
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

Localization of the photon on phase space

J. A. Brooke and F. E. Schroeck

J. Math. Phys. 37, 5958 (1996); http://dx.doi.org/10.1063/1.531759 (29 pages) | Cited 8 times

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We obtain phase space representations of the Poincaré group for zero mass particles of all helicities, including photons. A natural quantization scheme for massless particles arises, and a covariant phase space localization operator is found. © 1996 American Institute of Physics.
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03.65.Fd Algebraic methods
02.20.-a Group theory
11.30.Cp Lorentz and Poincaré invariance
11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries
14.70.Bh Photons

Multi‐periodic coherent states and the WKB exactness

Kazuyuki Fujii and Kunio Funahashi

J. Math. Phys. 37, 5987 (1996); http://dx.doi.org/10.1063/1.531760 (25 pages) | Cited 2 times

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We construct the path integral formula in terms of a ‘‘multi‐periodic’’ coherent state as an extension of the Nielsen–Rohrlich formula for spin. We make an exact calculation of the formula and show that, when a parameter corresponding to the magnitude of spin becomes large, the leading order term of the expansion coincides with the exact result. We also give an explicit correspondence between the trace formula in the multi‐periodic coherent state and the one in the ‘‘generalized’’ coherent state. © 1996 American Institute of Physics.
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02.30.Cj Measure and integration
71.15.-m Methods of electronic structure calculations

Hamiltonian structure of Dubrovin’s equation of associativity in 2‐d topological field theory

C. A. P. Galvão and Y. Nutku

J. Math. Phys. 37, 6012 (1996); http://dx.doi.org/10.1063/1.531761 (6 pages) | Cited 3 times

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A third order Monge‐Ampère type equation of associativity that Dubrovin has obtained in 2‐d topological field theory is formulated in terms of a variational principle subject to second class constraints. Using Dirac’s theory of constraints this degenerate Lagrangian system is cast into Hamiltonian form and the Hamiltonian operator is obtained from the Dirac bracket. There is a new type of Kac‐Moody algebra that corresponds to this Hamiltonian operator. In particular, it is not a W‐algebra. © 1996 American Institute of Physics.
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11.10.Ef Lagrangian and Hamiltonian approach
02.40.Pc General topology
02.10.-v Logic, set theory, and algebra
02.30.Xx Calculus of variations
02.30.Yy Control theory

Classical and quantum implications of the causality structure of two‐dimensional space–times with degenerate metrics

Jonathan Gratus and Robin W. Tucker

J. Math. Phys. 37, 6018 (1996); http://dx.doi.org/10.1063/1.531755 (15 pages) | Cited 3 times

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The causality structure of two‐dimensional manifolds with degenerate metrics is analyzed in terms of global solutions of the massless wave equation. Certain novel features emerge. Despite the absence of a traditional Lorentzian Cauchy surface on manifolds with a Euclidean domain, it is possible to uniquely determine a global solution (if it exists), satisfying well‐defined matching conditions at the degeneracy curve, from Cauchy data on certain spacelike curves in the Lorentzian region. In general, however, no global solution satisfying such matching conditions will be consistent with this data. Attention is drawn to a number of obstructions that arise prohibiting the construction of a bounded operator connecting asymptotic single particle states. The implications of these results for the existence of a unitary quantum field theory are discussed. © 1996 American Institute of Physics.
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02.40.Sf Manifolds and cell complexes
11.10.-z Field theory
03.65.Ge Solutions of wave equations: bound states
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Scattering in one dimension: The coupled Schrödinger equation, threshold behaviour and Levinson’s theorem

K. A. Kiers and W. van Dijk

J. Math. Phys. 37, 6033 (1996); http://dx.doi.org/10.1063/1.531762 (27 pages) | Cited 16 times

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We formulate scattering in one dimension due to the coupled Schrödinger equation in terms of the S matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson’s theorem is seen to have the form η(0)=π(nb+1/2n−1/2N), where η(0) is the phase of the S matrix at zero energy, nb the number of bound states with nonzero binding energy, n the number of half‐bound states, and N the number of coupled equations. In view of the effects due to the half‐bound states, the threshold behaviour of the scattering amplitudes is investigated in general, and is also illustrated by means of particular potential models. © 1996 American Institute of Physics.
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03.65.Nk Scattering theory
02.30.Em Potential theory
03.65.Ge Solutions of wave equations: bound states
11.55.-m S-matrix theory; analytic structure of amplitudes
02.10.-v Logic, set theory, and algebra
11.80.Gw Multichannel scattering

Special‐relativistic harmonic oscillator modeled by Klein–Gordon theory in anti‐de Sitter space

D. J. Navarro and J. Navarro‐Salas

J. Math. Phys. 37, 6060 (1996); http://dx.doi.org/10.1063/1.531763 (14 pages) | Cited 7 times

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It is shown that the one‐particle sector of the Klein–Gordon theory in the universal covering space of the anti‐de Sitter space (CAdS) can be interpreted, in a natural way, as a special‐relativistic oscillator in Minkowski space. The quantum wave functions have a significantly different behavior with respect to the nonrelativistic ones. The energy spectrum coincides, up to the ground state energy, with that of the nonrelativistic oscillator. The requirement of having the adequate nonrelativistic limit for the special‐relativistic oscillator theory turns out to be equivalent to the imposition of the Dirichlet‐type boundary condition at spatial infinity on CAdS Klein–Gordon functions. © 1996 American Institute of Physics.
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03.30.+p Special relativity
45.05.+x General theory of classical mechanics of discrete systems

Prepotential of N=2 SU(2) Yang–Mills gauge theory coupled with a massive matter multiplet

Yuji Ohta

J. Math. Phys. 37, 6074 (1996); http://dx.doi.org/10.1063/1.531764 (12 pages) | Cited 7 times

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We discuss the N=2 SU(2) Yang–Mills theory coupled with a massive matter in the weak coupling. In particular, we obtain the instanton expansion of its prepotential. Instanton contributions in the mass‐less limit are completely reproduced. We study also the double scaling limit of this massive theory and find that the prepotential with instanton corrections in the double scaling limit coincides with that of N=2 SU(2) Yang–Mills theory without matter. © 1996 American Institute of Physics.
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11.30.-j Symmetry and conservation laws
11.10.Jj Asymptotic problems and properties
11.15.-q Gauge field theories

Symmetry and history quantum theory: An analog of Wigner’s theorem

S. Schreckenberg

J. Math. Phys. 37, 6086 (1996); http://dx.doi.org/10.1063/1.531765 (20 pages) | Cited 10 times

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The basic ingredients of the ‘‘consistent histories’’ approach to quantum theory are a space UP of ‘‘history propositions’’ and a space D of ‘‘decoherence functionals.’’ In this article we consider such history quantum theories in the case where UP is given by the set of projectors P(V) on some Hilbert space V. We define the notion of a ‘‘physical symmetry of a history quantum theory’’ (PSHQT) and specify such objects exhaustively with the aid of an analog of Wigner’s theorem. In order to prove this theorem we investigate the structure of D, define the notion of an ‘‘elementary decoherence functional,’’ and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT ‘‘physically equivalent’’ and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously. © 1996 American Institute of Physics.
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03.65.Ta Foundations of quantum mechanics; measurement theory

Lie algebra cohomology and group structure of gauge theories

Hyun Seok Yang and Bum‐Hoon Lee

J. Math. Phys. 37, 6106 (1996); http://dx.doi.org/10.1063/1.531766 (15 pages) | Cited 5 times

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We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincaré duality for the Lie algebra cohomology of the infinite‐dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator Q° for the Lie algebra cohomology induced by BRST generator Q. We also point out an interesting duality relation—Poincaré duality—with respect to gauge anomalies and Wess–Zumino–Witten topological terms. We consider the consistent embedding of the BRST adjoint generator Q° into the relativistic phase space and identify the noncovariant symmetry recently discovered in QED with the BRST adjoint Nöther charge Q°. © 1996 American Institute of Physics.
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11.15.-q Gauge field theories
11.10.Lm Nonlinear or nonlocal theories and models
12.20.Ds Specific calculations
03.65.Pm Relativistic wave equations
11.30.Cp Lorentz and Poincaré invariance
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
02.20.Sv Lie algebras of Lie groups
02.40.Pc General topology

On q‐deformed supersymmetric classical mechanical models

L. P. Colatto and J. L. Matheus‐Valle

J. Math. Phys. 37, 6121 (1996); http://dx.doi.org/10.1063/1.531767 (9 pages) | Cited 5 times

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Based on the idea of quantum groups and para‐Grassmannian variables, we present a generalization of supersymmetric classical mechanics with a deformation parameter q=exp(2πi/k) dealing with the k=3 case. The coordinates of the q‐superspace are a commuting parameter t and a para‐Grassmannian variable θ, where θ3=0. The generator and covariant derivative are obtained, as well as the action for some possible superfields. © 1996 American Institute of Physics.
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03.65.Sq Semiclassical theories and applications
03.65.Fd Algebraic methods
45.05.+x General theory of classical mechanics of discrete systems
46.05.+b General theory of continuum mechanics of solids
02.20.-a Group theory
11.10.-z Field theory
11.30.Pb Supersymmetry

Inverse problem in nonstationary multidimensional medium

Boris M. Shevtsov

J. Math. Phys. 37, 6130 (1996); http://dx.doi.org/10.1063/1.531753 (9 pages) | Cited 1 time

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The problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered. The boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method. The integral‐differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method. The order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary. It was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium. The cause and character of the multidimensional inverse problem overdefiniteness are considered. The application of the proposed procedure for a statistical problem is discussed. © 1996 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
03.50.De Classical electromagnetism, Maxwell equations

A temperature and mass dependence of the linear Boltzmann collision operator from group theory point of view

Vladimir Saveliev

J. Math. Phys. 37, 6139 (1996); http://dx.doi.org/10.1063/1.531768 (13 pages) | Cited 4 times

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The Lie group of the transformations affecting the parameters of the linear Boltzmann collision operator such as temperature of background gas and ratio of masses of colliding particles and molecules is discovered. The group also describes the conservation laws for collisions and main symmetries of the collision operator. New algebraic properties of the collision operator are derived. Transformations acting on the variables and parameters and leaving the linear Boltzmann kinetic equation invariant are found. For the constant collision frequency the integral representation of solutions for nonuniform case in terms of the distribution function of particles drifting in a gas with zero temperature is deduced. The new exact relaxation solutions are obtained too. © 1996 American Institute of Physics.
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51.10.+y Kinetic and transport theory of gases
05.60.-k Transport processes
02.20.Sv Lie algebras of Lie groups
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

Rigorous estimates of small scales in turbulent flows

Peter Constantin, Charles R. Doering, and Edriss S. Titi

J. Math. Phys. 37, 6152 (1996); http://dx.doi.org/10.1063/1.531769 (5 pages) | Cited 5 times

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We derive rigorous bounds on the length scale of determining local averages (volume elements) for the 3‐D Navier‐Stokes Equations. These length scale estimates are related to Kolmogorov’s notion of a dissipation length scale in turbulent flows. © 1996 American Institute of Physics.
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47.27.E- Turbulence simulation and modeling
47.10.-g General theory in fluid dynamics

Explicit solutions of supersymmetric KP hierarchies: Supersolitons and solitinos

A. Ibort, L. Martínez Alonso, and E. Medina Reus

J. Math. Phys. 37, 6157 (1996); http://dx.doi.org/10.1063/1.531770 (16 pages) | Cited 12 times

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Wide classes of explicit solutions of the Manin‐Radul and Jacobian supersymmetric KP hierarchies are constructed by using line bundles over complex supercurves based on the Riemann sphere. Their construction extends several ideas of the standard KP theory, such as wave functions, ∂‐equations and τ‐functions. Thus, supersymmetric generalizations of N‐soliton solutions, including a new purely odd ‘‘solitino’’ solution, as well as rational solutions, are found and characterized. © 1996 American Institute of Physics.
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11.30.Pb Supersymmetry
11.10.Lm Nonlinear or nonlocal theories and models
14.80.Ly Supersymmetric partners of known particles

Lax–Nijenhuis operators for integrable systems

Y. Kosmann‐Schwarzbach and F. Magri

J. Math. Phys. 37, 6173 (1996); http://dx.doi.org/10.1063/1.531771 (25 pages) | Cited 7 times

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The relationship between Lax and bi‐Hamiltonian formulations of dynamical systems on finite‐ or infinite‐dimensional phase spaces is investigated. The Lax–Nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the Lenard recursion relations, while the converse holds for an operator with a simple spectrum. Explicit higher‐order Hamiltonian structures for the Toda system, a second Hamiltonian structure of the Euler equation for a rigid body in n‐dimensional space, and the quadratic Adler–Gelfand–Dickey structure for the KdV hierarchy are derived using the Lax–Nijenhuis equation. © 1996 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

A geometrical method towards first integrals for dynamical systems

Simon Labrunie and Robert Conte

J. Math. Phys. 37, 6198 (1996); http://dx.doi.org/10.1063/1.531772 (9 pages) | Cited 6 times

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We develop a method, based on Darboux’s and Liouville’s works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements’ forms. We apply it to three dynamical systems: Lotka–Volterra, Lorenz and Rikitake. © 1996 American Institute of Physics.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos
02.40.-k Geometry, differential geometry, and topology
02.30.Cj Measure and integration

The Kadomtsev–Petviashvili equation as a source of integrable model equations

Attilio Maccari

J. Math. Phys. 37, 6207 (1996); http://dx.doi.org/10.1063/1.531773 (6 pages) | Cited 47 times

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A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev–Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev–Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs. © 1996 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.30.Jr Partial differential equations

Wronskian solutions of the constrained Kadomtsev–Petviashvili hierarchy

Walter Oevel and Walter Strampp

J. Math. Phys. 37, 6213 (1996); http://dx.doi.org/10.1063/1.531788 (7 pages) | Cited 19 times

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The integrable Kadomtsev–Petviashvili (KP) hierarchy is compatible with generalized k‐constraints of the type (Lk)−=∑i qi∂−1xri. A large class of solutions—among them solitons—can be represented by Wronskian determinants of functions satisfying a set of linear equations. In this paper we shall obtain additional conditions for these functions imposed by the constraints. © 1996 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

Higher‐order Melnikov theory for adiabatic systems

Cristina Soto‐Treviño and Tasso J. Kaper

J. Math. Phys. 37, 6220 (1996); http://dx.doi.org/10.1063/1.531751 (30 pages) | Cited 8 times

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In this paper, we study adiabatic Hamiltonian systems including those subject to small‐amplitude forcing and damping. It is known that simple zeroes of the adiabatic Poincare–Arnold–Melnikov function imply the existence of primary intersection points of the stable and unstable manifolds of hyperbolic orbits. Here, we present an Nth‐order Melnikov function whose simple zeroes correspond to Nth‐order transverse intersection points and hence to N‐pulse homoclinic orbits. Using this function, it can be shown that N‐pulse homoclinic orbits arise in a plethora of adiabatic models, including systems with slowly varying potentials. The theory is illustrated on a damped Hamiltonian system with a slowly varying cubic potential. In addition, the Nth‐order adiabatic Melnikov function is useful for showing the existence of multi‐pulse resonant periodic orbits in the special class of slow, time‐periodic systems. © 1996 American Institute of Physics.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos

Probing quantum gravity through exactly soluble midi‐superspaces I

A. Ashtekar and M. Pierri

J. Math. Phys. 37, 6250 (1996); http://dx.doi.org/10.1063/1.531774 (21 pages) | Cited 47 times

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It is well‐known that the Einstein‐Rosen solutions to the 3+1‐ dimensional vacuum Einstein’s equations are in one to one correspondence with solutions of 2+1‐dimensional general relativity coupled to axi‐symmetric, zero rest mass scalar fields. We first re‐examine the quantization of this midi‐superspace paying special attention to the asymptotically flat boundary conditions and to certain functional analytic subtleties associated with regularization. We then use the resulting quantum theory to analyze several conceptual and technical issues of quantum gravity. © 1996 American Institute of Physics.
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04.60.-m Quantum gravity
11.30.Pb Supersymmetry
11.10.Jj Asymptotic problems and properties
11.15.-q Gauge field theories
04.20.Jb Exact solutions

Loop variables and holonomies for a class of conical space–times

V. B. Bezerra and P. S. Letelier

J. Math. Phys. 37, 6271 (1996); http://dx.doi.org/10.1063/1.531775 (12 pages) | Cited 4 times

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We compute the loop variables for a class of space–times with topological defects. In particular we compute these quantities for multiple moving cosmic strings and two plane topological defects crossed by a cosmic string, showing that these quantities are elements of the homogeneous Lorentz group. We also compute the loop variables for a multi‐chiral cone and we show that in the context of Einstein theory the loop variables are elements of the inhomogeneous Lorentz group, but in the context of Einstein‐Cartan theory they are elements of the homogeneous Lorentz group. © 1996 American Institute of Physics.
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04.20.Jb Exact solutions
11.27.+d Extended classical solutions; cosmic strings, domain walls, texture
02.40.-k Geometry, differential geometry, and topology
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)
02.40.Pc General topology
95.30.Sf Relativity and gravitation
02.20.-a Group theory
04.50.-h Higher-dimensional gravity and other theories of gravity
11.30.Rd Chiral symmetries
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