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Dec 1977

Volume 18, Issue 12, pp. 2243-2520

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A generalization of the Gel’fand–Levitan equation for the one‐dimensional Schrödinger equation

H. E. Moses

J. Math. Phys. 18, 2243 (1977); http://dx.doi.org/10.1063/1.523235 (8 pages) | Cited 8 times

Online Publication Date: 26 August 2008

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The Gel’fand–Levitan equation for the one‐dimensional Schrödinger equation is generalized to the case that the unperturbed Hamiltonian contains part of the scattering potential, this part being denoted by V0(x), and that the direct scattering problem has been solved for this Hamiltonian. Hence one knows the reflection coefficient b0(k), the point eigenvalues E0i, and the normalizations of the corresponding eigenfunctions C0i. We are given b1(k), E1i, C1i, which are the corresponding quantities for full potential V1(x) =V0(x)+ΔV (x). A Gel’fand–Levitan equation is set up in terms of b1(k)−b0(k) and the difference in measures for the discrete spectra for V0 and V1, respectively, from which ΔV can be found. One may regard the new algorithm as providing a means to modify a known potential to accommodate prescribed changes in the reflection coefficient and changes in the nature of the discrete spectrum. The generalization has applications to the Korteweg–de Vries equation. It is shown that a kind of ’’superposition’’ principle exists for solutions in that one can add a function of x and t to one solution and obtain a second solution. This principle can be used to separate the soliton part of a solution from the continuous spectrum part.
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02.30.Hq Ordinary differential equations
03.65.Db Functional analytical methods

Multi‐soliton‐like solutions to the Benjamin–Ono equation

R. I. Joseph

J. Math. Phys. 18, 2251 (1977); http://dx.doi.org/10.1063/1.523236 (8 pages) | Cited 10 times

Online Publication Date: 26 August 2008

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We outline a systematic method of obtaining particular solutions to the nonlinear, integrodifferential equation obtained by Benjamin and Ono in their study of the propagation of finite amplitude waves in fluids of great depth. These solutions have the property of asymptotically breaking up into a series of N spatially localized waves of permanent form; we loosely refer to these as ’’N‐soliton’’ solutions. Detailed results for the two‐soliton solution are explicitly given.
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02.30.Uu Integral transforms
02.30.Vv Operational calculus
47.10.-g General theory in fluid dynamics
02.60.Nm Integral and integrodifferential equations

Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups

J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus

J. Math. Phys. 18, 2259 (1977); http://dx.doi.org/10.1063/1.523237 (30 pages) | Cited 56 times

Online Publication Date: 26 August 2008

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An algorithm for classifying the closed connected subgroups S of a given Lie group G into conjugacy classes, presented in earlier papers, is further refined so as to provide us with ’’normalized’’ lists of representatives of subalgebra classes. The normalized lists contain the subgroup normalizer NorGS (NorGS is the largest subgroup of G for which S is an invariant subgroup) for each subgroup representative. The advantage of having normalized lists is that the problem of merging several different sublists (e.g., the lists of all subgroups of each maximal subgroup of G) into a single overall list becomes greatly simplified. The method is then applied to find all closed connected subgroups of the two de Sitter groups O(3,2) and O(4,1). The classification group in each case is the group of inner automorphisms.
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02.20.Rt Discrete subgroups of Lie groups

Finite and infinite measurement sequences in quantum mechanics and randomness: The Everett interpretation

Paul A. Benioff

J. Math. Phys. 18, 2289 (1977); http://dx.doi.org/10.1063/1.523238 (7 pages) | Cited 2 times

Online Publication Date: 26 August 2008

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The quantum mechanical description of both a finite and infinite number of measurement repetitions, as interactions between copies of an object system and a record system, are considered here. States describing the asymptotic situation of an infinite number of repetitions are seen to have some interesting properties. The main construction of the paper is the association of states to sequential tests for randomness. To each such test T and each positive integer m one can associate states ΘnTm and Θ∞Tm corresponding respectively to those length‐n and finite outcome sequences which pass test T at the significance level 2−m. Following the methods of Martin Löf, a universal sequential test V, which includes an infinity of sequential statistical tests for randomness, is given and the corresponding states ΘnVm and Θ∞Vm are discussed. Finally, a possible use of these states in the Everett interpretation of quantum mechanics is discussed.
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03.65.Ta Foundations of quantum mechanics; measurement theory

Vector spherical harmonics of the unit hyperboloid in Minkowski space

Levere Hostler

J. Math. Phys. 18, 2296 (1977); http://dx.doi.org/10.1063/1.523239 (12 pages) | Cited 2 times

Online Publication Date: 26 August 2008

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Vector fields in Minkowski space which are simultaneous eigenfunctions of the operators [Jx]2+[Jy]2 +[Jz]2, [Jz], and −(1/2) LμνLμν are investigated using special tensor methods which exploit the properties of the intrinsic gradient operator ∇ of the unit hyperboloid xμxμ=1. A convenient representation of the simultaneous eigenfunctions is provided by the use of Helmholtz’s theorem for the unit hyperboloid. The utility of this representation arises from the existence of intertwining relations such as Jμν∇=∇Lμν. An addition theorem for the solenoidal vector spherical harmonics of the unit hyperboloid is derived, and the Green’s function of Poisson’s equation on the unit hyperboloid is obtained.
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02.30.Tb Operator theory

Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation

Aubrey Truman

J. Math. Phys. 18, 2308 (1977); http://dx.doi.org/10.1063/1.523240 (8 pages) | Cited 22 times

Online Publication Date: 26 August 2008

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We consider the limiting case λ→0 of the Cauchy problem ∂uλ/∂t= (λ/2μ) ∇2xuλ +[V (x)/λ]uλ, uλ(x,0) =exp[−S0(x)/λ]T0(x); S0, T0 independent of λ, for both real and pure imaginary λ. We prove two new theorems relating the limiting solution of the above Cauchy problem to the corresponding equations of classical mechanics μ (d2x/dτ2)(τ) =−∇xV[x (τ)], τ∊ (0,t). These relationships include the physical result quantum mechanics → classical mechanics as h→0.
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02.30.Jr Partial differential equations
03.65.Ge Solutions of wave equations: bound states
05.60.-k Transport processes

Proof that the H− ion has only one bound state. Details and extension to finite nuclear mass

Robert Nyden Hill

J. Math. Phys. 18, 2316 (1977); http://dx.doi.org/10.1063/1.523241 (15 pages) | Cited 66 times

Online Publication Date: 26 August 2008

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It is rigorously demonstrated that the H− ion, treated in nonrelativistic approximation with Coulomb interactions only, has only one bound state for the electron to nucleus mass ratio less than 0.21010636. This extends earlier work which had proven the result in the fixed (infinite mass) nucleus approximation. The method used can, if desired, also be used to calculate rigorous lower bounds to the energies of those bound states of two electron atomic systems which do exist.
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03.65.Ge Solutions of wave equations: bound states
31.15.-p Calculations and mathematical techniques in atomic and molecular physics
31.15.V- Electron correlation calculations for atoms, ions and molecules

The generalized Langevin equation with Gaussian fluctuations

Ronald Forrest Fox

J. Math. Phys. 18, 2331 (1977); http://dx.doi.org/10.1063/1.523242 (5 pages) | Cited 43 times

Online Publication Date: 26 August 2008

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It is shown that all statistical properties of the generalized Langevin equation with Gaussian fluctuations are determined by a single, two‐point correlation function. The resulting description corresponds with a stationary, Gaussian, non‐Markovian process. Fokker–Planck‐like equations are discussed, and it is explained how they can lead one to the erroneous conclusion that the process is nonstationary, Gaussian, and Markovian.
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02.50.Ey Stochastic processes
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

N‐body quantum scattering theory in two Hilbert spaces. I. The basic equations

Colston Chandler and A. G. Gibson

J. Math. Phys. 18, 2336 (1977); http://dx.doi.org/10.1063/1.523243 (12 pages) | Cited 22 times

Online Publication Date: 26 August 2008

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Derivations are given for some transition and resolvent operator equations for multichannel quantum scattering with short‐range potentials. The basic difference between these and previous equations is that the unknown operators act only on the channel subspaces. This is made possible by utilizing, and extending, the two‐Hilbert‐space formulation previously given by the authors [in J. Math. Phys. 14, 1328 (1973)]. The equations in abstract form are of the Lippmann–Schwinger type, differing only in the appearance of certain injection operators from one Hilbert space to the other. When applied to multichannel quantum scattering, the abstract theory yields a new system of equations for the transition and resolvent operators. Uniqueness of the solution to the equations is proved.
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11.80.-m Relativistic scattering theory
11.80.Gw Multichannel scattering
11.80.Jy Many-body scattering and Faddeev equation

Analytic evaluation of an important integral in collision theory

John Detrich and Robert W. Conn

J. Math. Phys. 18, 2348 (1977); http://dx.doi.org/10.1063/1.523217 (4 pages) | Cited 3 times

Online Publication Date: 26 August 2008

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A commonly occurring integral in collision theory when the radial part of the potential has the form rne−αr is In;ll′(α;k,k′)  = F∞0dr rn+1 e−αrJl+1/2(kr) Jl′+1/2(k′r). Analytic results for this integral have been found for the most important cases, including those with negative values of n. This permits efficient evaluation of Born matrix elements required in many scattering theory applications based on perturbation methods.
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02.30.Uu Integral transforms
02.30.Vv Operational calculus
11.80.-m Relativistic scattering theory

Mirror planes in Newtonian stars with stratified flows

Lee Lindblom

J. Math. Phys. 18, 2352 (1977); http://dx.doi.org/10.1063/1.523218 (4 pages) | Cited 2 times

Online Publication Date: 26 August 2008

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This paper shows that a certain class of Newtonian stellar models must possess a plane of mirror symmetry. A corollary of this result is that static Newtonian stars must be spherical. The new features of the results given here are that: (a) The assumptions about the velocity distribution of the fluid are weaker than previous treatments and (b) the method of proof given here does not depend as strongly on the linearity of the gravitational field equations as the previously published treatments. Therefore, this proof may serve as a model for a general relativistic generalization of the mirror plane theorem.
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95.30.Lz Hydrodynamics
95.30.Sf Relativity and gravitation

Some solutions of stationary, axially‐symmetric gravitational field equations

Metin Gürses

J. Math. Phys. 18, 2356 (1977); http://dx.doi.org/10.1063/1.523219 (4 pages) | Cited 7 times

Online Publication Date: 26 August 2008

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Stationary, axially‐symmetric solutions of the gravitational field equations for vacuum, perfect fluid, and massless scalar field are considered. For the vacuum case, a similiar formulation to the one introduced by Ernst is presented by use of quarternions. Null dust solutions are found, and it is shown that they match with the van Stockum exterior solutions. An extension of the theorem by Eriş and Gürses is given which enables one to construct solutions to the gravitational field equations coupled with a charged dust and a massless scalar field from the solutions of the field equations coupled only with a charged dust.
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04.20.Jb Exact solutions

Universal singular functions in local field theory

Alfred C. T. Wu

J. Math. Phys. 18, 2360 (1977); http://dx.doi.org/10.1063/1.523220 (4 pages)

Online Publication Date: 26 August 2008

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The singularity structure of the universal singular functions in local field theory is simply seen by the stationary-phase method applied to the Lorentz group manifold.
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03.70.+k Theory of quantized fields
03.65.Fd Algebraic methods

KMS condition for stable states of infinite classical systems

M. Pulvirenti and G. Riela

J. Math. Phys. 18, 2364 (1977); http://dx.doi.org/10.1063/1.523221 (4 pages) | Cited 1 time

Online Publication Date: 26 August 2008

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Using an abstract algebraic approach, we obtain a new derivation of the KMS condition from a stability property of an infinite system via a classical version of the Tomita–Takesaki theorem.
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05.20.Gg Classical ensemble theory

Classical particles with spin. I. The WKBJ approximation

John Stachel and Jerzy Plebański

J. Math. Phys. 18, 2368 (1977); http://dx.doi.org/10.1063/1.523222 (7 pages) | Cited 3 times

Online Publication Date: 26 August 2008

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This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum‐mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single‐particle points of view is also elucidated.
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45.05.+x General theory of classical mechanics of discrete systems
11.90.+t Other topics in general theory of fields and particles (restricted to new topics in section 11)

Properties of causally continuous closed universes

Hirohisa Ishikawa

J. Math. Phys. 18, 2375 (1977); http://dx.doi.org/10.1063/1.523223 (2 pages) | Cited 2 times

Online Publication Date: 26 August 2008

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We consider the properties of causally continuous space–time with a closed spacelike hypersurface S, i.e., a closed universe. We show that a closed universe does not collide with other universes.
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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.)
04.20.Cv Fundamental problems and general formalism

Convergence acceleration technique for lattice sums arising in electronic‐structure studies of crystalline solids

Frank E. Harris

J. Math. Phys. 18, 2377 (1977); http://dx.doi.org/10.1063/1.523224 (5 pages)

Online Publication Date: 26 August 2008

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Slowly convergent lattice summations arise when ab initio quantum‐mechanical studies of electronic structure in crystalline solids are carried out by Fourier representation methods. Summations of this type are identified and discussed, and it is shown how a technique related to, but not identical with, that of Ewald can be used to accelerate their convergence. The presentation is illustrated with numerical examples.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.30.Lt Sequences, series, and summability

Casimir invariants and vector operators in simple and classical Lie algebras

Susumu Okubo

J. Math. Phys. 18, 2382 (1977); http://dx.doi.org/10.1063/1.523225 (13 pages) | Cited 56 times

Online Publication Date: 26 August 2008

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A method of computing eigenvalues of certain types of Casimir invariants has been developed for simple and classical Lie algebras. Especially these eigenvalues for algebras An, Bn, Cn, Dn, and G2 have been computed in closed terms. We also enumerate numbers and functional forms of all linearly independent vector operators in terms of generators in any irreducible representation of these algebras. Some polynomial identities among infinitesimal generators of these algebras are derived by means of the same technique.
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02.20.Sv Lie algebras of Lie groups

A new class of superalgebras and local gauge groups in superspace

Freydoon Mansouri

J. Math. Phys. 18, 2395 (1977); http://dx.doi.org/10.1063/1.523226 (2 pages) | Cited 3 times

Online Publication Date: 26 August 2008

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It is shown that there is a new class of superalgebras associated with a given Lie algebra or a superalgebra. The structure constants of the new algebras either vanish or else are directly related to those of the original algebra. The new algebraic structures provide a possible link between the local gauge groups constructed over superspace and those over ordinary space–time.
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02.20.Sv Lie algebras of Lie groups
11.15.-q Gauge field theories

The diffraction of sound pulses by a circular cylinder

Pocheng Chen and Yih−Hsing Pao

J. Math. Phys. 18, 2397 (1977); http://dx.doi.org/10.1063/1.523227 (10 pages)

Online Publication Date: 26 August 2008

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The diffraction of pulses in acoustic medium (scalar waves) by a circular cylinder is analyzed by applying the Cagniard method. Solutions for the incident, reflected, diffracted, and creeping pulses in the illuminated and shadow zones are all obtained by a unified approach. Numerical results are shown for the forward, backward, and side scattering of an incident pulse with a step or square time function.
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43.20.Fn Scattering of acoustic waves

Regularization of the Roy equations with a smooth cutoff

D. Atkinson, T. P. Pool, and H. A. Slim

J. Math. Phys. 18, 2407 (1977); http://dx.doi.org/10.1063/1.523228 (7 pages) | Cited 3 times

Online Publication Date: 26 August 2008

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The Roy equations for ππ scattering are combined with unitarity to give a nonlinear system of equations for the determination of the low‐energy amplitudes. A Hölder continuous interpolation between the input high‐energy absorptive parts and the output low‐energy absorptive parts is implemented; and the resultant singular equations are regularized by means of an effective inelastic N/D method. If the scattering lengths, the CDD parameters, and the high‐energy absorptive parts satisfy certain constraints, then there exists a locally unique solution of the system.
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11.80.Et Partial-wave analysis
11.80.-m Relativistic scattering theory
13.75.Lb Meson-meson interactions

Perturbation theory for Green’s functions as an effective mass formalism

F. Krause

J. Math. Phys. 18, 2414 (1977); http://dx.doi.org/10.1063/1.523229 (9 pages)

Online Publication Date: 26 August 2008

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A formal solution for Green’s functions of the type [∂2+m2+gV (x)]G (x,y) =δ (x−y) is presented which has the structure of an effective mass formalism. One first solves the free case [gV (x) =0] for given boundary conditions and then replaces the parameter m in the solution by a quantity depending on V (x). The rules for this replacement are given, a connection with the Baker–Campbell–Hausdorff formula is established, and it is shown how the formalism unites different perturbation and approximation schemes.
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02.30.Jr Partial differential equations
02.30.Sa Functional analysis

Massless quantum sine–Gordon equation in two space–time dimensions: Correlation inequalities and infinite volume limit

Yong Moon Park

J. Math. Phys. 18, 2423 (1977); http://dx.doi.org/10.1063/1.523230 (4 pages) | Cited 10 times

Online Publication Date: 26 August 2008

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We prove new correlation inequalities for the massive and massless quantum sine–Gordon equations. These results are then used to construct infinite volume limit theory for the massless (S–G)2 model that satisfies the Osterwalder–Schrader axioms. As consequences, infinite volume limit theories for the classical, neutral, statistical mechanical systems with two‐body Coulomb potentials and for the massive Thirring model exist.
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11.10.Cd Axiomatic approach

Klein–Gordon kinks with fourth order derivative self‐coupling

Antonio F. Rañada and Manuel F. Rañada

J. Math. Phys. 18, 2427 (1977); http://dx.doi.org/10.1063/1.523231 (5 pages) | Cited 4 times

Online Publication Date: 26 August 2008

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A Klein–Gordon field with a derivative fourth order self‐coupling is studied. It is shown that the kinks of the model form singlets, doublets, or triplets of electric charge, according to the values of the coupling constants.
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03.70.+k Theory of quantized fields

Time dependent canonical transformations and the symmetry‐equals‐invariant theorem

John R. Cary

J. Math. Phys. 18, 2432 (1977); http://dx.doi.org/10.1063/1.523232 (4 pages) | Cited 2 times

Online Publication Date: 26 August 2008

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Expressions for the remainder function of a time dependent infinitesimally generated canonical transformation have recently been found by Dewar, who considered the action of the transformation operators on Liouville’s equation. Here an alternate proof of the remainder function expression is given, based on the transformations of particle trajectories. Then, using this expression, a proof of the symmetry‐equals‐invariant theorem is given.
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45.05.+x General theory of classical mechanics of discrete systems
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