Volume 7, Issue 12, December 1966
Index of content:

Singular Integral Equations
View Description Hide DescriptionThe integral equationis shown to have simple solutions obtained by standard and elementary methods if h and K have appropriate analytic properties.

Singular Solutions of Certain Integral Equations
View Description Hide DescriptionA class of integral equations arising in some idealized plasma problems is discussed. While these do not have solutions in the space of square‐integrable functions, they do have such in an appropriate space of generalized functions. Explicit solutions are given in some special cases. It is then shown how these solutions can be used to approximate those for more general problems.

A Soluble Fermi‐Gas Model. Validity of Transformations of the Bogoliubov Type
View Description Hide DescriptionA linear soluble Fermi‐gas model is used to test the validity of perturbation theory and the criteria of instability obtained by applying transformations of the Bogoliubov type. The model Hamiltonian describing the interaction of the electrons in a simple band is.For a half‐filled band, the energies of the low‐lying states are calculated exactly, by using previous results concerning a linear chain of spins with interaction. It is shown that perturbation theory must be valid for −1 < ρ < 1, and, for values of ρ belonging to this range, the slope of the specific heat for T = 0 is calculated by direct application of Landau's theory. On the other hand, anomalous Hartree‐Fock states can be built for all values of ρ, by using transformations of the Bogoliubov type, and these states are more stable than the normal Hartree‐Fock state. Thus, it appears that the Bogoliubov method may lead to completely erroneous interpretations, when the coupling constants are small.

The Time‐Dependent Green's Function for Electromagnetic Waves in Moving Simple Media
View Description Hide DescriptionThis paper treats the problem of radiation from sources of arbitrary time dependence in a moving medium. The medium is assumed to be lossless, with permittivity ε and permeability μ, and to move with constant velocity v̄ with respect to a given inertial reference frame xyz. It is shown how the Maxwell‐Minkowskiequations for the electromagnetic fields in the moving medium can be integrated by means of a pair of vector and scalar potential functions analogous to those commonly used with stationary media. The wave equation associated with these potential functions is derived, and a scalar Green's function is defined to satisfy the same type of equation, with a delta‐function source term δ(r − r′) δ(t − t′), and the casuality condition. The solution for the Green's function is derived in closed form, by means of a Fourier integral method. The resulting Green's function is useful not only for calculating the fields from arbitrary sources in moving media, but also for its pedagogical value. It is simpler to understand the phenomenon of Cerenkov radiation using this method than it is from the conventional approach to the Cerenkov problem.

Propagation of Correlations in a Boltzmann Gas
View Description Hide DescriptionNew results are obtained on the propagation of correlations in a Boltzmann gas on the scale of the mean free path and the collisional time scale which appear to support M. Green's conjecture on this subject.

Peratization of the Logarithmically Singular Potential
View Description Hide DescriptionThe potential of the form is shown explicitly to be regularizable by an infinitesimal cutoff. It is also shown that, when a modified sense of the peratization technique is applied to the regulated scattering length, the correct answer is regained.

Algebraic Aspects of Regge Recurrences
View Description Hide DescriptionThe algebraic structure of Regge recurrences is examined, and its relation to symmetry schemes underlying various models is established. The relationship between the Spectrum Generating Algebras (SGA) and the noncompact subalgebras of the rotational states are discussed for the Coulomb potential, the three‐dimensional oscillator, the signature which appears in exchange potentials, as well as the U(6, 6) and the other suggested SGA of hadron physics.

Asymptotic Expansions of the Dirac Density Matrix
View Description Hide DescriptionAsymptotic expansions are derived for the density operatorfor small and large values of the relative distance x′ − x with a WKB expansion for bound‐state wavefunctions in a plane, one‐dimensional potential. In addition to some previously obtained corrections, the particle‐density expansion includes a number of steady and oscillating correction terms to the zero‐order Thomas‐Fermi density which are related to the more recently derived corrections of Payne and of Kohn and Sham. A convenient and accurate approximation to the density operator for all arguments is also obtained from the first two terms of the expansion for large relative distances. The expansions become inadequate in the vicinity of the classical turning point of the highest energy state, as illustrated by numerical comparisons with the exact‐density operator for the quadratic potential. A novel method is described for summing a finite series of oscillating terms such as occur in the density operator.

The Solution of the Nonrelativistic Quantum Scattering Problem without Exchange
View Description Hide DescriptionThe method of ``amplitude density functions'' is a new formalism which allows a scattering problem to be broken up into problems with weaker interaction potentials. These simpler problems may be solved separately and be ``added'' together to give the total solution. A numerical method is discussed which takes advantage of this property. The formulas are given for the use of this method in the solution of the one‐dimensional atom‐molecule collision and the e ^{+}−H collision. A numerical example is discussed.

Properties of a Causal Green's Function for the Bethe‐Salpeter Equation
View Description Hide DescriptionThe following properties of the free, but off‐mass‐shell, Bethe‐Salpeter causal Green's function, for two spinless bosons whose masses may differ, are investigated: (a) its symmetries, (b) its expression in terms of the common higher transcendental functions, and (c) its forms in the asymptotic region and near the origin. An energy‐spectral integral representation is also obtained.

A New Derivation of the Kinetic Equation for an Inhomogeneous Plasma
View Description Hide DescriptionThe kinetic equation for non‐uniform plasma is derived from the Liouville equation by the Prigogine‐Balescu diagram technique. The obtained equation is equivalent to the equation derived earlier by Balescu and this author, but has a little simpler form. The equation has a non‐Markoffian form and is valid in both stable and unstable cases and for long and short times. Its explicit form is obtained by using the Resibois' summation procedure. The limiting form of the equation for long times in the stable case is derived.

Global and Democratic Methods for Classifying N‐Particle States
View Description Hide DescriptionThe ``global method'' for describing N‐particle systems (which relies on the existence of a large invariance group of the total Hamiltonian for N noninteracting particles, the ``great group'', whose Lie algebra is generated by the ``grand angular momentumtensor''), is adapted to describe systems of identical particles by means of basis states with simple symmetry properties (with respect to permutations of the particles). We are led to define and study the concept of ``democracy'' among the particles, from which we obtain the ``democratic'' subgroups of the great group. The eigenvectors of a complete set of commuting observables, consisting essentially of Casimir operators of democratic subgroups, may furnish the desired basis. Unfortunately the scheme is seen to be sufficient only in the 3‐ and 4‐particle cases, which, however, are most important. The Appendix contains a discussion of the possible relativistic generalizations of the global method.

Hermitian Analyticity and S‐Matrix Singularity Structure
View Description Hide DescriptionIt is shown that the requirement of Hermitian analyticity can be used to determine the Riemann sheet structure of S‐matrix singularities. The examples of the square‐diagram Landau curve and of a single anomalous threshold are discussed for the (2 → 2)‐particle amplitude.

Representations of the Homogeneous Galilei Group in a Spherical Basis and the Asymptotic Behavior of Relativistic Special Functions
View Description Hide DescriptionThe irreducible unitary representations of the Galilei group G _{h} are obtained in the base lm〉 from their expression in the base q〉 (Mackey's representation). The faithful representations are then considered as asymptotic expressions for the representation matrices of the principal series of the Lorentz group.

Drum Shapes and Isospectral Graphs
View Description Hide DescriptionWe analyze the discrete drum model of Fisher and show that the information on the shape of a smooth drum contained in its low‐order spectral moments is, at best, the area, length of its boundary, and number and type of corners. We produce a counter example which proves that one cannot always hear the shape of a discrete drum.

Factorization Theorems
View Description Hide DescriptionA precise set of conditions is given for which the momentum correlation functions of quantum mechanical, weakly interacting, many‐particle systems can be factorized.

Relationship between Linear Response Theory and the Boltzmann Equation for a Weakly Interacting System of Electrons and Phonons
View Description Hide DescriptionVan Hove's weak coupling master equation and the factorization theorem of Kac are used to show that the transport coefficients derived from linear response theory are equivalent to those derived from the Boltzmann equation for a weakly interacting system of electrons and phonons.

Statistical Mechanics of Finite Systems: Asymptotic Expansions. I. Two Petit Canonical Ensembles—(N, V, T) and (N, p, T) Systems
View Description Hide DescriptionThe statistical mechanics of a finite system differs from that of an infinite system in details of calculation, possession of additional variables and modification of fundamental relations, i.e., altered thermodynamics. Results become size‐dependent, and ensemble‐dependent. As examples of the general situation for finite systems, two petit canonical ensembles are examined here in relation to the grand canonical ensemble. The full asymptotic expansion is obtained for a one‐phase region of one‐component systems for each ensemble; all terms can be calculated within a grand canonical ensemble formalism. An analysis is made of both size dependence and the modified thermodynamics resulting from various approximations. The method presented represents both an approach to the study the general problems of the statical mechanical treatment of finite systems as well as a practical device for calculation in particular problems of small systems.

A Class of Sum Rules with Application to Nondegenerate Perturbation Theory
View Description Hide DescriptionThe main results of this paper are: (1) The derivation of a class of sum rules. These rules are then applied to operators H(α), which depend on a parameter α and have eigenvaluesE_{n} (α). Writing A_{n} (α) ≡ H(α) − E_{n} (α), these rules give the matrix elements of ∂^{ N } A_{n} (α)/∂α^{ N } with respect to the eigenfunctions of H(α) in terms of matrix elements of lower derivatives and the eigenvaluesE_{n} (α). (2) An explicit series expression for the Nth‐order eigenvalue correction, due to a perturbation, directly in terms of the unperturbed eigenfunctions and eigenvalues. This expression seems a convenient one to use, since it does not involve eigenfunction corrections. An expression is also obtained for the eigenfunction corrections, but this is in less convenient form.

Minimal Potentials for Schrödinger Equation with Fixed Eigenvalue
View Description Hide DescriptionIt is shown that the potentialsolves the problem of minimizing , where V is such that the Schrödinger equation has an eigenvalueE, specified in advance, with a square‐integrable eigenfunction.