Volume 18, Issue 12, December 1977
Index of content:

A generalization of the Gel’fand–Levitan equation for the one‐dimensional Schrödinger equation
View Description Hide DescriptionThe 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 V _{0}(x), and that the direct scattering problem has been solved for this Hamiltonian. Hence one knows the reflection coefficientb _{0}(k), the point eigenvaluesE _{0i }, and the normalizations of the corresponding eigenfunctions C _{0i }. We are given b _{1}(k), E _{1i }, C _{1i }, which are the corresponding quantities for full potential V _{1}(x) =V _{0}(x)+ΔV (x). A Gel’fand–Levitan equation is set up in terms of b _{1}(k)−b _{0}(k) and the difference in measures for the discrete spectra for V _{0} and V _{1}, 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.

Multi‐soliton‐like solutions to the Benjamin–Ono equation
View Description Hide DescriptionWe 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.

Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups
View Description Hide DescriptionAn algorithm for classifying the closed connected subgroups S of a given Lie groupG 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 Nor_{ G } S (Nor_{ G } S 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.

Finite and infinite measurement sequences in quantum mechanics and randomness: The Everett interpretation
View Description Hide DescriptionThe 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 Θ_{ n } ^{ T m } and Θ^{∞T m } 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 Θ_{ n } ^{ V m } and Θ^{∞V m } are discussed. Finally, a possible use of these states in the Everett interpretation of quantum mechanics is discussed.

Vector spherical harmonics of the unit hyperboloid in Minkowski space
View Description Hide DescriptionVector fields in Minkowski space which are simultaneous eigenfunctions of the operators [J _{ x }]^{2}+[J _{ y }]^{2} +[J _{ z }]^{2}, [J _{ z }], and −(1/2) L ^{μν} L _{μν} are investigated using special tensor methods which exploit the properties of the intrinsic gradient operator ∇ of the unit hyperboloid x ^{μ} x _{μ}=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 J ^{μν}∇=∇L ^{μν}. 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.

Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation
View Description Hide DescriptionWe consider the limiting case λ→0 of the Cauchy problem ∂u _{λ}/∂t= (λ/2μ) ∇^{2} _{ x } u _{λ} +[V (x)/λ]u _{λ}, u _{λ}(x,0) =exp[−S _{0}(x)/λ]T _{0}(x); S _{0}, T _{0} 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 (d ^{2} x/dτ^{2})(τ) =−∇_{ x } V[x (τ)], τ∈ (0,t). These relationships include the physical result quantum mechanics → classical mechanics as h/→0.

Proof that the H^{−} ion has only one bound state. Details and extension to finite nuclear mass
View Description Hide DescriptionIt 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.

The generalized Langevin equation with Gaussian fluctuations
View Description Hide DescriptionIt 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 s t a t i o n a r y, G a u s s i a n, n o n‐M a r k o v i a n process. Fokker–Planck‐like equations are discussed, and it is explained how they can lead one to the erroneous conclusion that the process is n o n s t a t i o n a r y, G a u s s i a n, a n d M a r k o v i a n.

N‐body quantum scattering theory in two Hilbert spaces. I. The basic equations
View Description Hide DescriptionDerivations 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.

Analytic evaluation of an important integral in collision theory
View Description Hide DescriptionA commonly occurring integral in collision theory when the radial part of the potential has the form r ^{ n } e ^{−αr } is I _{ n;l l′}(α;k,k′) = F^{∞} _{0} d r r ^{ n+1} e ^{−αr } J _{ l+1/2}(k r) J _{ l′+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.

Mirror planes in Newtonian stars with stratified flows
View Description Hide DescriptionThis 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 fieldequations as the previously published treatments. Therefore, this proof may serve as a model for a general relativistic generalization of the mirror plane theorem.

Some solutions of stationary, axially‐symmetric gravitational field equations
View Description Hide DescriptionStationary, axially‐symmetric solutions of the gravitational fieldequations 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 fieldequations coupled with a charged dust and a massless scalar field from the solutions of the field equations coupled only with a charged dust.

Universal singular functions in local field theory
View Description Hide DescriptionThe singularity structure of the universal singular functions in local field theory is simply seen by the stationaryphase method applied to the Lorentz groupmanifold.

KMS condition for stable states of infinite classical systems
View Description Hide DescriptionUsing 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.

Classical particles with spin. I. The WKBJ approximation
View Description Hide DescriptionThis 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.

Properties of causally continuous closed universes
View Description Hide DescriptionWe 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.

Convergence acceleration technique for lattice sums arising in electronic‐structure studies of crystalline solids
View Description Hide DescriptionSlowly convergent lattice summations arise when a b i n i t i o 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.

Casimir invariants and vector operators in simple and classical Lie algebras
View Description Hide DescriptionA method of computing eigenvalues of certain types of Casimir invariants has been developed for simple and classical Lie algebras. Especially these eigenvalues for algebrasA _{ n }, B _{ n }, C _{ n }, D _{ n }, and G _{2} 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.

A new class of superalgebras and local gauge groups in superspace
View Description Hide DescriptionIt 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.

The diffraction of sound pulses by a circular cylinder
View Description Hide DescriptionThe 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.