Volume 16, Issue 5, May 1975
Index of content:

On the relaxation to quantum‐statistical equilibrium of the Wigner–Weisskopf atom in a one‐dimensional radiation field. VII. Emission in a finite system in the presence of an extra photon
View Description Hide DescriptionIn this paper, we present the exact solution to a problem previously unsolved in radiation theory: the emission of a two‐level atom in a (one‐dimensional) radiation field in the presence of an extra photon. The solution is obtained directly from the Schrödinger equation of the problem using techniques suggested by the work of Muskhelishvili on singular integral equations. The solution corresponding to a finite system as well as the one corresponding to a system infinite in extent are given, although our primary concern in this paper is the finite‐system problem. For a particular choice of initial condition, the probability at time t that the two‐level atom is in the excited state is found, and the effects of system size and choice of coupling function are studied numerically for a given coupling constant. Our results are compared with those obtained in an earlier paper of the series, wherein the spontaneous emission of a Wigner‐Weisskopf atom in a (one‐dimensional) field of radiation for a system finite in extent was studied. The physical effects calculated and the conclusions drawn from our comparative studies are all in accord with simple intuition regarding the problem. The paper concludes with some brief remarks on problems in radiation theory which are accessible to study given the methods laid down in this paper.

On the curvature dynamics for metric gravitational theories
View Description Hide DescriptionAny ’’metric gravitation theory’’ (including general relativity) is shown to determine transport equations for the connection and curvature of the Lorentz frame bundle P _{4} defined by the metric g. Observers are generally defined as curves in P _{4} which project down to timelike trajectories in space–time. The transport of curvature along an observer trajectory is then given by a Lorentz Lie algebra‐valued current composed of an internal and external part. Einstein’s equations are shown to define one part of the self‐dual limit of the usual Yang–Mills gauge equations, here called a particular form of curvature dynamics. As a consequence, the Yang–Mills‐like energy–momentum tensor, introduced for the Lorentz connection, vanishes identically under Einstein’s vacuum conditions.

Conformal invariance and Hamilton Jacobi theory for dissipative systems
View Description Hide DescriptionFor certain dissipative systems, a comparison can be made between the Hamilton–Jacobi theory and the conformal invariance of action theory. The two concepts are not identical, but the conformal action theory covers the Hamilton–Jacobi theory.

Weyl quantization of anharmonic oscillators
View Description Hide DescriptionIt is shown that polynomial self‐interactions appearing repeatedly in the mathematical physics and chemical physics literature for the case of one degree of freedom can be treated in the formalism suggested by Weyl long ago, as was the case for the harmonic oscillator. New properties of the Schrödinger wavefunctions are derived, and appropriate schemes of approximation for the eigenvalue problem arise naturally in a nonperturbative way.

An integral equation for the Gel’fand–Levitan kernel in terms of the scattering potential in the one‐dimensional case
View Description Hide DescriptionAn integral equation for the Gel’fand–Levitan kernel is given in terms of the scattering potential. This integral equation may be regarded as complementary to the Gel’fand–Levitan equation which is an integral equation for the kernel in terms of the Fourier transform of the reflection coefficient.

Variable dimensionality in the group‐theoretic prediction of configuration mixings for doubly‐excited helium
View Description Hide DescriptionVariable dimensionality (= D) is used to interpret recent group‐theoretic predictions of configuration mixings in doubly‐excited helium. Calculated 2s ^{2}:2p ^{2} ^{1} S mixings agree with the group theory over the range 1 ⩽ D ⩽ ∞. General results for D = 2 predicted mixings are given and energies of states within the N = 2 atomic shell confirm predicted level orderings. The D = 1 model atom is described exactly by the group theory, with quantum numbers and a selection rule characterizing the stability of Coulomb matrix elements as D → 1. The exact D = 1 results have a physical interpretation in approximate autoionization and energy selection rules for Rydberg series at D = 3.

The inverse scattering transform: Semi‐infinite interval
View Description Hide DescriptionCertain nonlinear evolution equations can be solved on the semi‐infinite interval by the method of inverse scattering. These equations are a subset of those which can be solved on the full interval. The equations have even dispersion relations when linearized, and are subject to appropriate homogeneous boundary conditions at the origin.

On the transformation from random Schrödinger Hamiltonians to random Hamiltonian matrices
View Description Hide DescriptionIn this article we consider the effects on ensembles of random Hamiltonian matrices when certain restrictions are imposed on the potentials of the corresponding Schrödinger Hamiltonians. In particular, we investigate the validity of the assumption which is usually made when ensembles of random matrices are used to predict statistical properties of energy levels of complex systems, namely that the Hamiltonian matrix elements are statistically independent.

Lattice approximation of the (λφ^{4} − μφ)_{3} field theory in a finite volume
View Description Hide DescriptionFor space–time regions with finite volumes, the lattice cutoff unnormalized Schwinger functions converge to the (unnormalized) Schwinger functions as the lattice cutoff is removed.

Linked cluster expansions in the density fluctuation formulation of the many‐body problem
View Description Hide DescriptionA perturbation theory is developed for the logarithm of the normalization integral (or partition function) of an N‐body system which is either a Bose liquid in its ground state or a classical fluid in the canonical ensemble. The perturbation in the former is an n‐body factor in the ground statewavefunction and in the latter is an n‐body potential. The normalization function serves as a generating function for the cumulants (or static correlation functions) of the density fluctuation operator ρ_{k}. The expansions of the perturbed partition function and correlation functions are shown to be linked cluster expansions involving the correlation functions in the unperturbed system; each term in these expansions is manifestly of the proper order in N. Several approximations involving truncations of the cumulants and/or resummation of part of the terms in the linked cluster expansions are discussed.

A note on coherent state representations of Lie groups
View Description Hide DescriptionThe analyticity properties of coherent states for a semisimple Lie group are discussed. It is shown that they lead naturally to a classical ’’phase space realization’’ of the group.

The de Donder coordinate condition and minimal class 1 space–time
View Description Hide DescriptionBy means of immersion techniques a set of ’’adapted coordinates’’ are introduced as preferred coordinates for class 1 space–time. It is proved that the necessary and sufficient condition for the adapted coordinates to be harmonic coordinates is that class 1 space–time be a minimal variety. Some interesting features of the embedding approach to

Bifurcate nondiverging null hypersurfaces and trapped surfaces
View Description Hide DescriptionAn empty spacetime containing a bifurcate nondiverging null hypersurface is investigated, and conditions are given that are necessary and sufficient for the existence of trapped surfaces near this hypersurface to its future. These conditions involve a topological requirement that the two‐surface of bifurcation be compact and an inequality that must be satisfied by the characteristic data for this spacetime—the metric of the two‐surface of bifurcation and an arbitrary function given on this surface that is shown to be related to angular momentum. The existence of a bifurcate Killing horizon in this spacetime is established. Finally, a Kerr spacetime containing bifurcate Killing horizons is examined, and results pertaining to the existence of trapped surfaces near these horizons to their futures are obtained. These results involve the parameters representing mass and angular momentum per unit mass.

Variational principles, variational identities, and supervariational principles for wavefunctions
View Description Hide DescriptionWe develop variational principles and variational identities for bound state and continuum wavefunctions in a general context, paying particular attention to the proper choice of defining equations and boundary conditions which will lead to unique and unambiguous wavefunctions even when these functions are complex. Any functional, such as a matrix element, calculated with such a variationally determined wavefunction, will also be accurate to second order in the error of the starting choice. This provides, therefore, an alternative procedure for getting variational estimates of matrix elements to the one that already exists in the literature and we establish the equivalence of the two. Of even more interest is the possibility which now seems open of going beyond the variational principle and generating ’’supervariational’’ estimates of wavefunctions and matrix elements which are good to better than second order. We also give a simple prescription for the construction of variational identities for wavefunctions, that is, identities which lead readily to variational principles and, more significantly, might well serve as a starting point for the development of variational bounds.

Pyramidal composition rules for Wronskians upon Wronskians
View Description Hide DescriptionWe give composition rules for Wronskians which have Wronskians as arguments. These pyramids of Wronskians are shown to reduce to products of Wronskians of different order. Many symmetric and antisymmetric regroupings of functions are then possible. The arithmetic of Wronskians can efficiently be reduced to the application of the of the LSZ formalism.

Extension of the Case formulas to L _{ p }. Application to half and full space problems
View Description Hide DescriptionThe singular eigenfunction expansions originally applied by Case to solutions of the transport equation are extended from the space of Hölder‐continuous functions to the function spaces X _{ p } = {f‖μf (μ) ‐ L _{ p }}, where the expansions are now to be interpreted in the X _{ p } norm. The spectral family for the transport operator is then obtained explicitly, and is used to solve transport problems with X _{ p } sources and incident distributions.

Global existence of solutions to the Cauchy problem for time‐dependent Hartree equations
View Description Hide DescriptionThe existence of global solutions to the Cauchy problem for time‐dependent Hartree equations for N electrons is established. The solution is shown to have a uniformly bounded H ^{1}(R^{3}) norm and to satisfy an estimate of the form ∥ ψ (t) ∥_{ H } _{ 2 } ⩽ c exp(k t). It is shown that ’’negative energy’’ solutions do not converge uniformly to zero as t → ∞.

The classical nonlinear oscillator and the coherent state
View Description Hide DescriptionThe coherent state constructed out of quantum oscillator states is employed to develop a method for solving the classical nonlinear oscillator problem. The perturbation solution of the Duffing oscillator is used to illustrate the method and to obtain the result of the classical procedure.

An iterative method for solving a two‐point boundary‐value problem
View Description Hide DescriptionThe purpose of this paper is to study the existence, uniqueness, and method of construction of a nonnegative solution as well as the question of criticality for a two‐point boundary‐value problem arising in the transport process of n different types of particles in a rod of finite length subjecting incident fluxes and internal source. A recursion formula is derived for the calculation of the maximal and the minimal solution which are the respective limits of a monotonically nonincreasing sequence and a monotonically nondecreasing sequence. The behavior of these sequences leads to a characterization for the criticality question of the transport problem. It is shown under some physically reasonable conditions that the minimal and maximal solutions coincide so that it leads to a uniqueness theorem.

O (4,2) symmetry and the classical Kepler problem
View Description Hide DescriptionThe spectrum generating algebra associated with the negative energy motions of a classical dynamical system, namely the Kepler problem, has been systematically studied with the aid of the Poisson brackets. The canonical map between our realization and that of an earlier analysis by Barut and Bornzin has been established.