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Jan 1978

Volume 19, Issue 1, pp. 1-348

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On the first approximation of the K‐harmonics method

J. A. Castilho Alcarás

J. Math. Phys. 19, 1 (1978); http://dx.doi.org/10.1063/1.523538 (4 pages) | Cited 6 times

Online Publication Date: 11 August 2008

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Formulas for the calculation of two‐body central interactions in the first approximation of the K‐harmonics method are presented. These formulas are exact and account for diagonal and off‐diagonal matrix elements as well.
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21.65.-f Nuclear matter
02.30.Gp Special functions

Self‐avoiding random walks: Some exactly soluble cases

Deepak Dhar

J. Math. Phys. 19, 5 (1978); http://dx.doi.org/10.1063/1.523515 (7 pages) | Cited 86 times

Online Publication Date: 11 August 2008

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We use the exact renormalization group equations to determine the asymptotic behavior of long self‐avoiding random walks on some pseudolattices. The lattices considered are the truncated 3‐simplex, the truncated 4‐simplex, and the modified rectangular lattices. The total number of random walks Cn, the number of polygons Pn of perimeter n, and the mean square end to end distance 〈R2n〉 are assumed to be asymptotically proportional to μnnγ−1, μnnα−3, and n2ν respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant μ, and the critical exponents λ, α, ν are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents γ, α, and ν.
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64.60.Cn Order-disorder transformations
05.70.Fh Phase transitions: general studies

On projective symmetries of dynamical systems

Toshihiro Iwai

J. Math. Phys. 19, 12 (1978); http://dx.doi.org/10.1063/1.523529 (7 pages) | Cited 1 time

Online Publication Date: 11 August 2008

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The present paper is concerned with symmetry transformations of a dynamical system defined on the tangent bundle of a Riemannian manifold. Of present interest are infinitesimal symmetry transformations of the vector field which defines the dynamical system on the tangent bundle. It is known that a class of such transformations entails infinitesimal projective transformations leaving the vector field invariant. Symmetry algebras formed by such projective transformations are studied. It is shown which dynamical systems admit large symmetry algebras. As a result, two kinds of dynamical systems are determined, which have the base Riemannian manifolds of constant curvature with dimensions n?4. The systems are generalizations of the classical harmonic oscillator and Kepler problem usually considered in Euclidean spaces. First integrals quadratic in the velocities are obtained, which are also generalizations of the well‐known quadratic integrals for the above classical systems.
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45.05.+x General theory of classical mechanics of discrete systems

’’IST‐solvable’’ nonlinear evolution equations and existence—An extension of Lax’s method

I. Miodek

J. Math. Phys. 19, 19 (1978); http://dx.doi.org/10.1063/1.523537 (13 pages) | Cited 14 times

Online Publication Date: 11 August 2008

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We present here a new and easy method, a natural extension of Lax’s method, for obtaining general ’’IST‐solvable’’ nonlinear evolution equations. These are evolution equations for the potential function(s), v, of a Hamiltonian, H, when the logarithmic t derivatives of H’s inverse scattering data are given by a t‐dependent ratio of entire functions of E, Ω (t,E). Here E is the energy variable and Ω is the ’’dispersion relation’’ of Abowitz, Kaup, Newell, and Segur (AKNS). We pose the question of existence of the evolution equation’s solution. This question is answered completely in the one‐dimensional Schrödinger case (first example). In a second example we derive the evolution equation for an n×n matrix generalization of the Zakharov–Shabat–AKNS equation. Our method displays the central role of analyticity in E in the IST method as a whole.
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03.65.Db Functional analytical methods
03.65.Nk Scattering theory

Dense electron‐gas response at any degeneracy

C. Gouedard and C. Deutsch

J. Math. Phys. 19, 32 (1978); http://dx.doi.org/10.1063/1.523508 (7 pages) | Cited 63 times

Online Publication Date: 11 August 2008

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See Also: Erratum

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An exact expression for the linear response function of the dense electron gas valid at any temperature is worked out in the ring (RPA) approximation. The T=0 and T=∞ limits reproduce the already known results. It is used to explain the longitudinal oscillations and the screening around a test charge. The latter is either Thomas–Fermi‐like or Friedel‐like according to the values of the parameters.
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05.30.Fk Fermion systems and electron gas
72.15.Eb Electrical and thermal conduction in crystalline metals and alloys

Dynamics and phase transitions for a continuous system of quantum particles in a box

Guy A. Battle

J. Math. Phys. 19, 39 (1978); http://dx.doi.org/10.1063/1.523512 (13 pages)

Online Publication Date: 11 August 2008

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A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one‐to‐one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.
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05.30.-d Quantum statistical mechanics

Inverse Gaussian transforms: General properties and application to Slater‐type orbitals with noninteger and integer n in the coordinate and momentum representations

William J. Taylor

J. Math. Phys. 19, 52 (1978); http://dx.doi.org/10.1063/1.523513 (7 pages) | Cited 9 times

Online Publication Date: 11 August 2008

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See Also: Erratum

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The use of Gaussian‐type orbitals (GTO) facilitates the evaluation of the multicenter integrals encountered in quantum chemistry by reducing all integrals of more than two centers to two‐center integrals. On the other hand Slater‐type orbitals (STO), while leading to more time‐consuming integral evaluations, provide a better approximation to variationally determined atomic orbitals. Thus, for a basis set of given size, STO’s generally give better accuracy than GTO’s. Kikuchi proposed the representation of STO’s as integral Gaussian transforms, or in effect by continuous expansions in GTO’s, and Shavitt, Karplus, and Kern have applied this technique to the evaluation of multicenter integrals over STO’s. If these procedures are to be extended, it is desirable to develop a more systematic approach to the representation of a given basis function, ψ (r), as a Gaussian transform, ψ (r) =G[f (t);r] =F∞0f (t)exp(−r2t) dt; what this reduces to is the problem of calculating the inverse Gaussian transform, f (t) =G−1[ψ (r);t]. In the present investigation it is pointed out that f (t) =L−1[ψ (s1/2);t], where L−1 represents the inverse Laplace transformation. On this basis conditions on ψ (r) necessary for the existence of a unique continuous Gaussian inverse, f (t), are formulated, and general rules for the manipulation of inverse Gaussian transforms are developed. Finally, the formulas for the inverse Gaussian transforms of STO’s obtained previously by Kikuchi and Wright are generalized to noninteger principal quantum number, and angle‐dependent STO’s, in both the coordinate and momentum representations.
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02.30.Uu Integral transforms
02.30.Vv Operational calculus
31.15.A- Ab initio calculations

Nontranslationally covariant currents and associated symmetry generators

W. D. Garber and H. Reeh

J. Math. Phys. 19, 59 (1978); http://dx.doi.org/10.1063/1.523514 (8 pages) | Cited 9 times

Online Publication Date: 11 August 2008

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Within the framework of axiomatic field theory, the general case of a translationally noncovariant conserved local current is investigated. It is shown that the associated symmetry does not change the particle number nor the mass or the momentum of one‐particle states. There is an integer N such that the N‐fold commutators of the generator with the momentum as well as with the mass operator vanish.
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11.10.Cd Axiomatic approach

On the wave‐mechanical representation of a Bose‐like oscillator

Y. Ohnuki and S. Kamefuchi

J. Math. Phys. 19, 67 (1978); http://dx.doi.org/10.1063/1.523516 (12 pages) | Cited 22 times

Online Publication Date: 11 August 2008

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A detailed study is made of the wave‐mechanical representation of a one‐dimensional Bose‐like oscillator whose canonical variables satisfy the general commutation relations first proposed by Wigner. The eigenvalue problems of the momentum and Hamiltonian operators are completely solved, and this is made possible only when wavefunctions in general are allowed to be hyperfunctions. The equivalence between the wave‐ and matrix‐mechanical representations is thereby established for any value of c (a characteristic parameter of the theory), contrary to the conclusion reached previously by Yang. It is also found that for the case −1/2<c<0 or 0<c<1/2 there exist two classes of eigenfunctions that are mutually separated by a superselection rule.
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03.65.Ge Solutions of wave equations: bound states

The three‐dimensional convolution of reduced Bessel functions and other functions of physical interest

E. Filter and E. O. Steinborn

J. Math. Phys. 19, 79 (1978); http://dx.doi.org/10.1063/1.523517 (6 pages) | Cited 61 times

Online Publication Date: 11 August 2008

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A method for evaluating convolution integrals over rather general functions is suggested, based on the analytical evaluation of convolution integrals over functions BMν,L(r) = (2/π)1/2rL+νKν (r) YML(ϑ,ϕ), which are products of modified Bessel functions of the second kind Kν(r), regular solid spherical harmonics rLYML(ϑ,ϕ), and powers rν.
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02.30.Gp Special functions

A generalized prolongation structure and the Bäcklund transformation of the anticommuting massive Thirring model

H. C. Morris

J. Math. Phys. 19, 85 (1978); http://dx.doi.org/10.1063/1.523518 (3 pages) | Cited 5 times

Online Publication Date: 11 August 2008

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The prolongation structure method of Wahlquist and Estabrook is generalized to Grassmann algebra valued differential forms and used to determine a Bäcklund transformation for the equations of the anticommuting massive Thirring model.
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02.10.De Algebraic structures and number theory
11.10.Cd Axiomatic approach

Eigenvalues of S⋅Π for spins 1/2, 1, and 3/2

D. L. Weaver

J. Math. Phys. 19, 88 (1978); http://dx.doi.org/10.1063/1.523519 (4 pages) | Cited 6 times

Online Publication Date: 11 August 2008

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The eigenvalues of the matrix operator S⋅Π for a constant magnetic field are derived in a parallel way for spins 1/2, 1, and 3/2 using only the algebra of the spin matrices and the commutation relations of the components of Π.
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03.65.Fd Algebraic methods

Spaces of positive and negative frequency solutions of field equations in curved space–times. II. The massive vector field equations in static space–times

Carlos Moreno

J. Math. Phys. 19, 92 (1978); http://dx.doi.org/10.1063/1.523520 (8 pages) | Cited 2 times

Online Publication Date: 11 August 2008

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The space–times considered in this article are static, Vn×R, with compact space‐section manifolds without boundary, Vn, and such that the trajectories of the Killing vector field are geodesics. For the physical field of spin 1 and mass m≳0 in these space–times, field equations are solved in any adapted atlas, by the one‐parameter groups of unitary operators generated by scalar and vector Hamiltonians, i−1Tj−1, j=0,1, in Sobolev spaces Hjl−1(Vn) ×Hjl 1(Vn), lϵR. Hilbert spaces of positive energy solutions of field equations, as well as those of reduced solutions and their canonical symplectic and complex structures, are determined. The existence and the uniqueness of Lichnerowicz’s (1−1) current on space–time are established, and the corresponding frequency‐solution Hilbert spaces are constructed. Within the framework of Segal, a definition of quantum field operators is given, leading to the postulated commutator for the physical field concerned.
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03.70.+k Theory of quantized fields

A variational derivation of the Bach–Lanczos identity

John R. Ray

J. Math. Phys. 19, 100 (1978); http://dx.doi.org/10.1063/1.523525 (3 pages) | Cited 4 times

Online Publication Date: 11 August 2008

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A discussion of a modified Hilbert variational principle is presented. The Bach–Lanczos identity is then derived from this variational principle.
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04.20.Fy Canonical formalism, Lagrangians, and variational principles
02.30.Xx Calculus of variations
02.30.Yy Control theory

Uniqueness connection between charge conjugation and statistics

C. L. Hammer and B. DeFacio

J. Math. Phys. 19, 103 (1978); http://dx.doi.org/10.1063/1.523526 (6 pages)

Online Publication Date: 11 August 2008

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The charge conjugation properties of bilinear quantum field theories are examined in considerable detail. It is shown that the connection between charge conjugation and statistics is unique. The relation between spin and statistics for a large class of these theories and the statistics of unusual fields such as Faddeev–Popov ghost fields and Gupta’s regularizing fields with negative norm are discussed.
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11.10.-z Field theory
11.80.-m Relativistic scattering theory

Theory of vibrations of coated, thermopiezoelectric laminae

M. Cengiz Dökmeci

J. Math. Phys. 19, 109 (1978); http://dx.doi.org/10.1063/1.523527 (18 pages) | Cited 16 times

Online Publication Date: 11 August 2008

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This study presents a theory for dynamic problems of coated laminae in which there is coupling between mechanical and electrical as well as thermal fields. The laminae is coated completely with perfectly conducting electrodes on both its faces, and it may comprise any number of bonded layers, each with a distinct but uniform thickness, curvature and electromechanical properties. First, a generalized variational theorem is derived so as to describe the complete set of the fundamental equations of thermopiezoelectricity. Next, by the use of this theorem, a system of two‐dimensional, approximate governing equations of the coated laminae is constructed for the case when the mechanical displacement, electric potential, and temperature fields vary linearly across the laminae thickness. The effects of elastic stiffnesses of, and the interactions between, layers of the laminae and its electrodes are all taken into account. Also, the uniqueness of the governing equations is examined, and a theorem which includes the conditions sufficient for the uniqueness is given.
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77.65.-j Piezoelectricity and electromechanical effects
73.90.+f Other topics in electronic structure and electrical properties of surfaces, interfaces, thin films, and low-dimensional structures (Restricted to new topics in section 73)
77.70.+a Pyroelectric and electrocaloric effects

Fluctuation theories and Gaussian stochastic processes

Ronald Forrest Fox

J. Math. Phys. 19, 127 (1978); http://dx.doi.org/10.1063/1.523528 (4 pages) | Cited 6 times

Online Publication Date: 11 August 2008

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The theory of fluctuations for systems near equilibrium has given rise to two developments which generalize the theory in two distinct ways. One of these developments is focused on the theory of fluctuations far from equilibrium where the dynamics is nonlinear. The other development has focused on extending the class of fluctuating forces to include forces with non‐delta function correlations. The near equilibrium theory corresponds with the theory of stationary, Gaussian, Markov processes; the nonlinear, far from equilibrium theory corresponds to the theory of nonstationary, Gaussian, Markov processes; and the non‐delta function, force correlation theory corresponds to the theory of stationary, Gaussian, non‐Markovian processes in one form, and to the theory of nonstationary, Gaussian, non‐Markovian processes in another form. The common feature found in all these theories is Gaussianness.
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02.50.Ey Stochastic processes
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Transforming Gaussians into Wannier functions

Gregory H. Wannier

J. Math. Phys. 19, 131 (1978); http://dx.doi.org/10.1063/1.523530 (4 pages) | Cited 17 times

Online Publication Date: 11 August 2008

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It is clear a priori that equal Gaussian functions, spread over a lattice, can be transformed into Wannier functions. The transformation is carried out here analytically in one dimension, with the help of the theory of theta functions. The results confirm and illustrate the properties commonly assigned to these functions, with one startling exception.
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71.15.-m Methods of electronic structure calculations

A derivation of the virial expansion with application to Euclidean quantum field theory

R. Menikoff and D. H. Sharp

J. Math. Phys. 19, 135 (1978); http://dx.doi.org/10.1063/1.523531 (16 pages) | Cited 11 times

Online Publication Date: 11 August 2008

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In this paper we give a derivation of the virial expansion and some of its generalizations. Our derivation is based on the generating functional which defines a representation of the density operator ρ (x) in a nonrelativistic local current algebra. The virial expansion results from solving a functional differential equation for this quantity. We exploit the well‐known analogy between quantum field theory and classical statistical mechanics to explore the use of the virial expansion in Euclidean quantum field theory. Specifically, we show that the virial expansion can be used to derive Feynman’s rules and to provide a perturbation expansion about a static ultralocal model. The latter is worked out in detail in the case of a free neutral scalar model, and outlined in the case of a λϕ4 model.
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03.70.+k Theory of quantized fields

Multiple steady states in a simple reaction–diffusion model with Michaelis–Menten (first‐order Hinshelwood–Langmuir) saturation law: The limit of large separation in the two diffusion constants

J. L. Ibañez and M. G. Velarde

J. Math. Phys. 19, 151 (1978); http://dx.doi.org/10.1063/1.523532 (6 pages) | Cited 9 times

Online Publication Date: 11 August 2008

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The admissible multiple nonuniform steady states of a model bimolecular autocatalytic reaction–diffusion system with Michaelis–Menten (first‐order Hinshelwood–Langmuir) saturation law are constructed in the case of large scale separation in the two diffusion constants. Both the Dirichlet and the Neumann problems are discussed in a one‐dimensional geometry, and the corresponding bifurcation pictures are given.
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02.30.Jr Partial differential equations
31.70.Hq Time-dependent phenomena: excitation and relaxation processes, and reaction rates

A characterization of Lorentz transformations

A. Lenard

J. Math. Phys. 19, 157 (1978); http://dx.doi.org/10.1063/1.523533 (1 page) | Cited 1 time

Online Publication Date: 11 August 2008

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If a one‐to‐one correspondence of Minkowski space–time onto itself is such that timelike lines, and only timelike lines, map onto timelike lines, then the correspondence is an inhomogeneous Lorentz transformation combined with a dilation.
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03.30.+p Special relativity

Expanding shearfree spatially homogeneous universes with a nonsynchronous time coordinate and anisotropy of the universe

A. J. Fennelly

J. Math. Phys. 19, 158 (1978); http://dx.doi.org/10.1063/1.523534 (6 pages) | Cited 2 times

Online Publication Date: 11 August 2008

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The apparent isotropy of the microwave background radiation on all angular scales and regions thus far observed is usually accepted as evidence of the isotropy of the universe. Current applications of the Einstein theory of gravitation in cosmology couples all rotation or peculiar velocity to anisotropy in the background radiation. Rotation in particular is coupled to shear. These are based on a number of results that apply to spatially homogeneous cosmologies with a synchronous time coordinate. While the models allow the past of our universe to be rather interesting, they constrain the present to be quite boring. We introduce a nonsynchronous time coordinate and show that the present could allow more interesting fluid motions, including rotation and peculiar velocity in spite of the microwave background’s isotropy. We use the formalism to construct rotating Friedman universes and study observations in them. Almost all other cosmological data are consistent with a rotation of the universe.
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98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

A new approach to the eigenvalues of the Gel’fand invariants for the unitary, orthogonal, and symplectic groups

S. A. Edwards

J. Math. Phys. 19, 164 (1978); http://dx.doi.org/10.1063/1.523535 (4 pages) | Cited 23 times

Online Publication Date: 11 August 2008

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The expressions for the Gel’fand invariants (Casimir operators) of U(n), O(n), and Sp(n) in terms of the IR labels are derived by relating them to the trace of a suitably defined operator Pk. The method has unity and simplicity, since trace (Pk) can be related directly to the IR labels using the Weyl dimension function—without having to determine the eigenvectors of P.
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02.20.Rt Discrete subgroups of Lie groups
03.65.Fd Algebraic methods

On the self‐induced transparency effect of the three‐wave resonance process

S. C. Chiu

J. Math. Phys. 19, 168 (1978); http://dx.doi.org/10.1063/1.523536 (9 pages) | Cited 12 times

Online Publication Date: 11 August 2008

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The self‐induced transparency effect of the three‐wave resonance is studied by the inverse scattering method. By transforming to the characteristic coordinates of the background wave, the inverse‐scattering theory becomes greatly simplified. With the presence of a constant background wave, the three‐wave process is dispersive, and the solitons and continuum behave in a different way from those of spatially bounded wavepackets. The continuum decays away, depositing energy to the background. The solitons have velocities which are amplitude dependent.
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42.50.Md Optical transient phenomena: quantum beats, photon echo, free-induction decay, dephasings and revivals, optical nutation, and self-induced transparency
42.25.Kb Coherence

Perturbation theory for Coulomb scattering

J. Zorbas

J. Math. Phys. 19, 177 (1978); http://dx.doi.org/10.1063/1.523522 (10 pages) | Cited 9 times

Online Publication Date: 11 August 2008

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Perturbation series for the ’’renormalized’’ complex‐energy distorted plane waves (RCEW’s) are defined and shown to converge to the series expansions for the physical distorted plane waves (PW’s) for two‐particle scattering via Coulomb‐like potentials V (x) =e1e2‖x‖−1+Vs(x), where Vs ∊ L2 has compact support. A perturbation series for the ’’renormalized’’ half‐off‐shell T matrix is defined and shown to converge to the series expansion for the pure Coulomb physical T matrix.
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11.55.-m S-matrix theory; analytic structure of amplitudes
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