Volume 16, Issue 4, April 1975
Index of content:

Theory of Regge poles for 1/r ^{2} potentials. I
View Description Hide DescriptionIn the following series of papers we give a detailed study of the theory of Regge poles for 1/r ^{2} potentials with the behavior r ^{2} V (r) = −V _{0} at r = 0 and r ^{2} V (r) = −V _{2} at r = ∞. We give a complete description of the distribution of the Regge poles in the λ plane, which is cut from −V _{0} ^{1/2} to V _{0} ^{1/2} and study the behavior of the pole trajectories. We find that the high energy limit of the Regge poles is controlled by the parameter p = (λ^{2}−V _{0})^{1/2} while at low energies the relevant parameter is q = (λ^{2}−V _{2})^{1/2}. This means that the point λ = 0 for the case of Yukawa potentials corresponds here to the point q = 0. We also find that the Regge trajectories λ (E) may have branch points of the square root type at finite, in general complex, values of E at which points the pole passes the origin λ = 0. We further find that the kinematic singularity of the S matrix at k = 0 is more complicated than it is for Yukawa potentials and is here characterized by the Floquet parameter ν (λ,k) associated with the Schrödinger equation. We illustrate these and other results with some

Theory of Regge poles for 1/r ^{2} potentials. II. An exactly solvable example at zero energy
View Description Hide DescriptionWe give the exact solution of the Schrödinger equation at zero energy and derive an expression in closed form for the Regge poles for a particular 1/r ^{2} potential with the behavior r ^{2} V (r) = −V _{0} at r = 0 and r ^{2} V (r) = −V _{2} at r = ∞. We give detailed results on the properties and distribution of the Regge poles in the λ plane and find them to be in agreement with the predictions of a previous paper in this series.

Theory of Regge poles for 1/r ^{2} potentials. III. An exact solution of Schrödinger’s equation for arbitrary l and E
View Description Hide DescriptionWe give the exact solution of the radial Schrödinger equation and derive the S matrix for arbitrary energy and angular momentum for a particular 1/r ^{2} potential with the behavior r ^{2} V (r) = −V _{0} at r = 0 and r ^{2} V (r) = −V _{2} at r = ∞. We obtain an expression for the Regge poles and study their properties and distribution at low energy. The present results are in agreement with those obtained in a previous paper in this series.

Double coset analysis for symmetry adapting Nth rank tensors of U (n) to its unitary subgroups
View Description Hide DescriptionRepresentations of the unitary group U (n) symmetry adapted to the subgroup sequencesU (n) ^{ U } ^{(n } ^{ j } ^{)} _{ U } _{(} _{ i } _{ n)} U (_{ i } n _{ j }) are considered using double coset decomposition. The matrix elements of the double coset representatives are related to identical coefficients developed

Second order error in variational calculation of matrix elements
View Description Hide DescriptionThe second order error δF ^{(2)}, made by employing the Schwartz−Dalgarno−Delves variational principle for the diagonal matrix element φ^{†} Wφ of an arbitrary Hermitian operator W, is examined in the case that φ is the bound ground state eigenfunction of some given Hamiltonian H. This variational principle characteristically involves not only a trial estimate φ_{ t } of φ, but also a trial estimate L _{ t }(φ_{ t }) L _{ t }(φ_{ t }) of a well−defined but generally not exactly known auxiliary function L (φ). It has previously been shown that, for certain special choices of a trial Hamiltonian H _{ t }, the trial L _{ t }(φ_{ t }) can be found from a minimum principle. The present work finds that for these same special H _{ t } it is possible to express δF ^{(2)} in comparatively simple closed form, depending only on known quantities, so that δF ^{(2)} should be calculable when the system described by H is not too complicated. However, these results for δF ^{(2)} are obtained on the assumption that L _{ t }(φ_{ t }) is known essentially exactly for any given φ_{ t }; the practical utility of the formulas derived still must be tested, therefore. If L _{ t }(φ_{ t }) can be determined to this necessary accuracy, one expects that combining the computed δF ^{(2)} with the usual variational estimate of φ^{†} Wφ will be a significant improvement over the usual variational estimate alone. Under the same circumstances, when W is a positive definite operator, the expression for δF ^{(2)} can provide nonrigorous but nonetheless potentially useful second order (variational) upper and lower bounds on the exact φ^{†} Wφ.

A note on the unified Dirac−von Neumann formulation of quantum mechanics
View Description Hide DescriptionIt is demonstrated that the mathematical model of a rigged Hilbert space is ideally suited for obtaining a unified Dirac−von Neumann formulation of quantum mechanics. It is shown that the eigenbras of an observable A can be interpreted as weak derivatives of certain functionals associated with the resolution of identity E _{ u }, u ‐ (−∞, ∞), associated with A.

Temporally inhomogeneous scattering theory for modified wave operators
View Description Hide DescriptionA theorem of Alsholm and Kato, which gives existence of modified wave operators for a large class of long range potentials, is extended to include time dependent potentials. It is then shown that these temporally inhomogeneous modified wave operators W _{ D±}(S) vary continuously, in some sense, on the potentials. This result is new for both time dependent and time independent potentials. In addition, part of the nonuniqueness problem of modified wave operators is resolved by noting that the modified wave operators

Studies in the Kerr−Newman metric
View Description Hide DescriptionThe Kerr−Newman metric is analyzed according to the null tetrad formalism. The components of the Weyl and the Ricci tensors are calculated and these tensors are then projected on a suitable null−tetrad basis. The spin coefficients of Newman and Penrose are also calculated. These results are applied to obtain the equations of gravitational and neutrino perturbations in the Kerr−Newman metric.

Mathematical aspects of kinetic model equations for binary gas mixtures
View Description Hide DescriptionA system of integrodifferential equations, which has a structure similar to the Boltzmann equations for a binary gas mixture and which qualitatively describes wave propagation, is investigated. The Oppenheim model is used and a linear initial−value problem is considered. The initial−value problem is shown to be well set mathematically with certain specifications on the initial distribution functions. Justification is made for the use of Fourier−Laplace transforms. A discussion is made of the dispersion relation and its analytic continuation. The roots σ (k) of the dispersion relation are shown to lie in three distinct regions of the σ plane: the hydrodynamic region, the semihydrodynamic region, and the rarefied region. It is established that the roots σ (k) are bounded by −1 + δ < Reσ ⩽ 0 under the assumption of plane−wave solutions which implies that the system is stable and that plane waves cease to exist if Reσ ⩽ −1 + δ.

Vestigial effects of singular potentials in diffusion theory and quantum mechanics
View Description Hide DescriptionRepulsive singular potentials of the form λV (x) =λ‖x−c‖^{−α}, λ≳0, in the Feynman−Kac integral are studied as a function of α. For α≳2 such potentials completely suppress the contribution to the integral from paths that reach the singularity, and thus, unavoidably, certain vestiges of the potential remain even after the coefficient λ↓0. For 2⩾α⩾1 careful definition by means of suitable counterterms at the point of singularity (similar in spirit to renormalization counter terms in field theory) can lead to complete elimination of effects of the potential as λ↓0. For α<1 no residual effects of the potential exist as λ↓0. In order to prove these results we rely on the theory of stochastic processes using, in particular, local time and stochastic differential equations. These results established for the Feynman−Kac integral conform with those known in the theory of differential equations. In fact, a variety of vestigial effects can arise from suitable choices of counter terms, and these correspond in a natural way to various self−adjoint extensions of the formal differential operator.

Matrix elements of the generators of I U (n) and I O (n) and respective deformations to U (n,1) and O (n,1)
View Description Hide DescriptionThe unitary continuous representations of U (n,1) and O (n,1) are discussed from the point of view of deformation of I U (n) and I O (n). It is shown that there are two general ways of writing the matrix elements of the infinitesimal generators of the groups U (n,1) and O (n,1). The first one is to write them as either pure real or pure imaginary. The second one is to write them as complex. We show how these different ways are related to each other.

Killing inequalities for relativistically rotating fluids
View Description Hide DescriptionFor rigidly rotating fluids in general relativity, it is shown that the angular momentum density is everywhere positive. This result depends on a global inequality satisfied by the Killing scalars. The inequality follows, via the Hopf theorem, from an elliptic equation (essentially one of the field equations) on the scalars. A derivation of the field equations in terms of the manifold of Killing orbits is presented. Possible generalizations of the result to systems with differential rotation or interior event horizons are discussed.

Lattice Green’s function for the body−centered cubic lattice
View Description Hide DescriptionWe have shown that the lattice Green’s function at an arbitrary site with nearest neighbor interactions for the body−centered cubic lattice is expressed as a finite sum of products of the complete elliptic integrals of the first and the second kinds with real values of moduli for the entire range of energy.

Convergence of Padé approximants using the solution of linear functional equations
View Description Hide DescriptionWe prove that a projection of the solutions to a linear functional equation of the Fredholm type with a compact kernel, projected into the Cini−Fubini subspaces, converge strongly to the solution in the whole space. Here either the whole sequence converges for all nonsingular points of the functional equation with at most one exceptional point, or by selecting at most two infinite subsequences we can obtain convergence for all nonsingular points. We then prove that the diagonal Padé approximants to the inner product of the solution with another element converge. For certain kernels of trace class, the numerator and denominator separately converge. As applications of these results, we prove the pointwise convergence of the Padé approximants to a wide class of meromorphic functions. We also prove the convergence, for decent potentials, of the Padé approximants to the scattering amplitudes for nonrelativistic quantum mechanical scattering problems. The numerators and denominators of the Padé approximants to the partial wave scattering amplitudes for single signed potentials converge separately to entire

Experimental uncertainties in the problem of the unitarity equation
View Description Hide DescriptionIn this article the question of how experimental uncertainties affect the construction of the scattering amplitude from the differential cross section and unitarity at a fixed energy is examined. It is shown that in most cases in which the solution can be found by the method of Newton and Martin, the problem is ’’well−posed’’ in the sense that the solution depends continuously on the data. A new proof is given of the fact that if the differential cross section is nearly constant and small enough, there is a unique solution of the problem. Some estimates for the scattering amplitude

Conformally flat solutions of the Einstein−Maxwell equations for null electromagnetic fields
View Description Hide DescriptionThe spin coefficient formalism of Newman and Penrose is employed to obtain a direct derivation of the most general conformally flat solution of the source−free Einstein−Maxwell equations for null electromagnetic fields.

Algebraic approach of the infrared−problem for external currents
View Description Hide DescriptionThe infrared problem for external currents is shown to be a consequence of the nonexistence of a particle number in the correct representation. A

NUT−like generalization of axisymmetric gravitational fields
View Description Hide DescriptionThe complex potential formulation of the axisymmetric problem discussed by Ernst enables us to construct new solutions from a given one, by multiplying the corresponding potential by a unit complex number. This rotation introduces naturally the NUT parameter in the metric. The generalized Kerr, Weyl, and Tomimatsu−Sato solutions are explicity constructed.

Nonexistence of dissipative structure solutions to Volterra many−species models
View Description Hide DescriptionSubject to boundary conditions of practical interest, the only temporally periodic solutions that may be admitted by a generic system of Volterra n−species reaction−diffusion equations are spatially uniform solutions, and thus dissipative structures are precluded as solutions to Volterra n−species models.

Symmetry of ensembles of maximum entropy
View Description Hide DescriptionThere can be only o n e maximally random ensemble in a given convex−closed family of ensembles, because the mixing of several ensembles increases entropy. Hence, if the family is acted on by a group which does not modify randomness(entropy), the thermodynamic ensemble is i n v a r i a n t. This is clear only over a finite−dimensional Hilbert space, prior to thermodynamic limits. Hence, in this situation, strictly spontaneous breakdown of symmetry is impossible.