Volume 10, Issue 2, February 1969
Index of content:

Relationships among Generalized Phase‐Space Distributions
View Description Hide DescriptionThe generalized phase‐space distributions, including the Wigner distribution, are presented in terms of expected values of generating operators. A generalization of the Weyl correspondence is obtained to provide expressions for generalized Wigner equivalents. Finally, rather simple relationships are obtained connecting the generalized phase‐space distributions to the Wigner distribution; and similar relationships are obtained for the generalized Wigner equivalents. In particular, it appears that, among the class considered, there is no reason to use any distribution other than the Wigner for performing any calculations.

Symmetry of the Ground Level of a Hamiltonian
View Description Hide DescriptionA general connection between nodelessness and symmetry of a function is pointed out. It is proved that a real nodeless energy eigenfunction with energy E has a nonzero part which also is an eigenfunction with energy E and which under coordinate transformations has the full symmetry of the Hamiltonian. This result can be applied to many systems of physical interest for which the ground‐state energy eigenfunction is known from the nature of the Hamiltonian to be nodeless. Simple counterexamples are given however, to show that not all Hamiltonians have a nodeless ground‐state energy eigenfunction.

Wave Propagation and Other Spectral Problems in Kinetic Theory
View Description Hide DescriptionA rigorous discussion of the spectrum of the linear Boltzmann equation and kinetic models is presented. Particular attention is given to plane‐wave propagation for a general class of kinetic models. These models in general have a velocity‐dependent collision frequency ν(ξ). The main results of this paper concern the relationship between the complex wavenumber and complex frequency for a plane wave. It is shown that the question of analyticity of this relation is reduced to considering ν in the neighborhood of infinity. Specifically, if,the relationship is not analytic. Otherwise, analyticity is obtained. (Although not specifically considered here, analyticity is closely connected to the convergence of the Chapman—Enskog procedure.) In a general discussion it is shown how the question of analyticity is closely connected with (i) the continuous spectrum of the underlying operator, (ii) the behavior of solutions at large distances from boundaries, and (iii) the nature of the cutoff in an intermolecular interaction.

Closed‐Form Solution of the Differential Equation Subject to the Initial Condition P(x, y, t = 0) = Φ(x, y)
View Description Hide DescriptionA closed‐form solution of the differential equation,subject to the initial condition P(x, y, t = 0) = Φ(x, y), is presented.

Inner and Restriction Multiplicity for Classical Groups
View Description Hide DescriptionFor the classical compact Lie groupsG, a formula for the multiplicity of weights (called inner multiplicity) is given. This formula relates the inner multiplicity of a group G to the inner multiplicity of a naturally embedded subgroup G′. For the SU(n) groups the formula can be brought into a particularly simple form—namely, a sum over Kronecker symbols—by choosing the group SU(2) for G′. The multiplicity of irreducible representations of a subgroup G′ into which an irreducible representation of a group G decomposes if G is restricted to G′—called restriction multiplicity of G with respect to G′—is related to the inner multiplicity of the group G.

Theory of the Propagation of the Plane‐Wave Disturbances in a Distribution of Thermal Neutrons
View Description Hide DescriptionThe propagation of plane‐wave disturbances in a neutron gas in thermal equilibrium with a moderator is studied using the linearized Boltzmann equation. The eigenvalue spectrum of the appropriate Boltzmanntransport operator is investigated for both polycrystalline and noncrystalline media, and several necessary conditions for the existence of a point spectrum (and hence for the existence of plane‐wave propagation) are discussed. Techniques are presented which allow the solution of various boundary‐value problems using this spectral representation.

Symmetry of the Two‐Dimensional Hydrogen Atom
View Description Hide DescriptionA paradox has arisen from some recent treatments of accidental degeneracy which claim that, for three degrees of freedom, SU(3) should be a universal symmetry group. Such conclusions are in disagreement both with experimentally observed spectra and with the generally accepted solutions of Schrödinger's equation. The discrepancy occurs in the transition between classical and quantum‐mechanical formulations of the problems, and illustrates the care necessary in forming quantum‐mechanical operators from classical expressions. The hydrogen atom in parabolic coordinates in two dimensions, for which the traditional treatment of Fock, extended by Alliluev, requires the symmetry group O(3), is a case for which the newer methods of Fradkin, Mukunda, Dulock, and others require SU(2). Although these groups are only slightly different, SU(2) fails to be the ``universal'' symmetry group on account of the multiple‐valuedness of the parabolic representation. This conclusion extends a result of Han and Stehle: that, for rather similar reasons, SU(2) cannot be the classical symmetry group for the two‐dimensional hydrogen atom.

Elementary Solutions of the Reduced Three‐Dimensional Transport Equation
View Description Hide DescriptionBy treating one of the space dimensions exactly and approximating the other two by the exp (−i B·r) assumption, which is suggested by asymptotic transport theory, it is possible to reduce the three‐dimensional transportequation to an equation that is of one‐dimensional form and that still contains details of the complete three‐dimensional angular distribution. In this paper we develop the method of elementary solutions for the reduced transportequation in the case of time‐independent, monoenergetic neutron transport in homogeneous media with isotropic scattering. The spectrum of the transport operator consists of a pair of discrete points if B ^{2} is sufficiently small and a continuum which occupies a two‐dimensional region in the complex spectral plane. The eigenfunctions possess full‐range and half‐range orthogonality and completeness properties, which are proved via the solution of two‐dimensional integral equations using the theory of boundary‐value problems for generalized analytic functions. As applications we solve the Green's function for an infinite homogeneous prism and the albedo operator for a semi‐infinite homogeneous prism. Also discussed are possible generalizations of the method to more complicated forms of the reduced transportequation.

De Sitter Symmetric Field Theory. I. One‐Particle Theory
View Description Hide DescriptionThe formal structure and the free‐particle solutions of the field equations (S_{μν}p_{ν} + mγ_{μ})ψ = p _{μ}ψ, derived recently by the author [Nuovo Cimento 51A, 864 (1967)] for realizations of the inhomogeneous de Sitter group are discussed. The enveloping algebra of the group is developed, and the covariance of the field equations under the five‐dimensional rotations C, P, and T is proved. Bhabha's representation of the matrices γ_{μ} is completed. Observables, expectation, values and the scalar product are defined, and classical conservation laws are derived. The field equations are derived from a variational principle for the usual Lagrangian density under a certain restriction. The free‐particle solutions of the field equations are obtained in the canonical and the extreme relativistic representations. The connections between the wavefunctions in these representations, and also in the Foldy‐Wouthuysen representation, are derived.

Inversion Problem with Separable Potentials
View Description Hide DescriptionThe problem of finding a potential, expressible as the sum of a finite number of separable terms, to fit a given phase shift at all energies is solved quite generally for nonrelativistic scattering within a single channel. The most general solution is found, and the necessary restrictions on the phase shift and the role of bound states are studied. The minimum number of separable terms needed to fit a given phase shift in a given channel is found to be determined, normally, by,where x _{1} and x _{2} range through all positive energies, and [y] is the largest integer less than or equal to y. Our method differs from that of Kh. Chadan [Nuovo Cimento 10, 892 (1958); 47, 510 (1967)] who has solved what is, in essence, the same problem, in that it involves spherical Bessel transforms throughout, rather than the use at each stage of a representation determined by the solution of a previous stage.

Classification and Complete Reducibility of Representations of Compact Inhomogeneous Groups and Algebras
View Description Hide DescriptionThe representations of compact inhomogeneous groups are classified according to which type of operators are taken as diagonal. The question of complete reducibility of some of the different classes is discussed.

Crossing and Unitarity in a Multichannel Static Model. II
View Description Hide DescriptionConstraints on the solution of the unitarity equation for a two‐channel scattering problem arising from the requirements of crossing for the inelastic amplitude are shown to imply the relations imposed by crossing under SU(2) for the elastic amplitudes in the limit of vanishing coupling to the inelastic channel. This result is extended to the three‐channel case, where at least two distinct classes of internal symmetry crossing relations can exist: the unitarity equations alone simulate the SU(2) case, but not that associated with higher symmetry groups.

Perturbation Theory of Nonlinear Boundary‐Value Problems
View Description Hide DescriptionA systematic perturbation theory is presented for the analysis of nonlinear problems. The lowest‐order result is just that obtained by linearizing the problem, and the higher‐order terms are the solutions of inhomogeneous linear problems. The essential feature of the method is the procedure for avoiding secular terms, which is based on the Lindstedt‐Poincaré technique employed in celestial mechanics. The method is applied to the following nonlinear boundary value problems: (1) temperature distribution due to a nonlinear heat source or sink; (2) self‐sustained oscillations of a system with infinitely many degrees of freedom; (3) forced vibrations of a ``string'' with a nonlinear restoring force; (4) superconductivity in a body of arbitrary shape with external magnetic field; (5) superconductivity in an infinite film with parallel magnetic field; (6) comparison of solutions of the Hartree, Fock, and Schrödinger equations for the helium atom. The results in each case are different both qualitatively and quantitatively from those of the linear theory.

Experimental Confirmation of the Applicability of the Fokker‐Planck Equation to a Nonlinear Oscillator
View Description Hide DescriptionA traditional derivation of the Fokker‐Planck equation is examined with emphasis on assumptions, especially sufficient inequalities between excitation and response characteristic times in a ``real'' system, i.e., a system with finite correlation time of the excitation. Comparison with experiment for an electronic ``cubic spring'' oscillator gives good agreement in both second and fourth moments of the response.

Generalization of Green's Theorem
View Description Hide DescriptionFor a system of field equations which is derivable from a Lagrangian whose density is (i) homogeneous quadratic in the first derivatives of the field variables y_{A,μ} and (ii) homogeneous of degree n in the undifferentiated field variables y_{A} , one has the identity,where M^{A} (y, z) is the first‐order change in the field equationsL^{A} (y) = 0 under the mapping y_{A} → y_{A} + z_{A} . The specific example of general relativity is discussed.