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Volume 22, Issue 8, August 1981

Some properties of the operator algebra generated by Hodge’s star and the exterior derivative
View Description Hide DescriptionSome properties of the operator algebra generated by Hodge’s star and the exterior derivative are established and in particular it is shown that viewed as an algebra over its center, this algebra is four‐dimensional.

Fermionic coherent states in a Fock superspace
View Description Hide DescriptionFermionic coherent states (FCS) are constructed via the Weyl supergroup of isometries of a Fock superspace. Their properties are derived and the connection with the pseudomechanics formalism is pointed out.

Canonical transformations relating the oscillator and Coulomb problems and their relevance for collective motions
View Description Hide DescriptionThe present paper can be viewed from two standpoints. The first is that it derives the canonical transformation that takes the Hamiltonian of the Coulomb problem (in the Fock–Bargmann formulation) into that of the harmonic oscillator, while transforming the angular momenta of both problems into each other. The second is the one in which the solution of the previous problem is required if we wish to find the canonical transformation relating microscopic and macroscopic collective models, where the former is derived from a system of A particles moving in two dimensions and interacting through harmonic oscillator forces. The canonical transformation shows the existence of a U(3) symmetry group in the microscopic collective model corresponding to that of the three‐dimensional oscillator which is the Hamiltonian of the macroscopic collective model. The importance of this result rests on the fact that had the motion of the particles taken place in the physical three‐dimensional space, rather than the hypothetical two‐dimensional one discussed here, the symmetry group would have been U(6) rather than U(3). Thus, the group theoretical structure of an s‐d boson picture or, equivalently, of a generalized Bohr–Mottelson approach, is present implicitly in an A‐body system interacting through harmonic oscillator forces.

Representations and Clebsch–Gordan coefficients of Z‐metacyclic groups
View Description Hide DescriptionWe investigate finite groups of the form Z_{ m }sZ_{ n } and give all irreducible representations and Clebsch–Gordan coefficients in analytic form. Two subclasses are considered which seem to be inportant for applications: the M‐metacyclic groups which are important for spin systems, and the K‐metacyclic groups which are the smallest finite groups that have an irreducible representation of dimension p−1, where p is prime.

Finite subgroups of SU(3)
View Description Hide DescriptionWe present a new class of finite subgroups of SU(3) of the form Z_{ m } s Z_{ n } (semidirect product). We also apply the methods used to investigate semidirect products to the known SU(3) subgroups Δ(3n ^{2}) and Δ(6n ^{2}) and give analytic formulae for representations (characters) and Clebsch–Gordan coefficients.

Semiunitary projective representations of the complete Galilei group
View Description Hide DescriptionA systematic study of the semiunitary projective representations of the Galilei group including reflections is presented. They are found by means of the semiunitary representations of a representation group.

A unified treatment of the representation functions of SO(n,1), SO(n+1), and ISO(n)
View Description Hide DescriptionAn explicit expression is obtained for all the representation functions of SO(n,1), SO(n+1), and ISO(n). It is found that the representation functions of SO(n,1) and SO(n+1) are basically expressible as hypergeometric functions _{2} F _{1} with arguments 1‐e ^{−2} ^{ζ} and 1‐e ^{2} ^{ iδ}, respectively, for n≳2, multiplied by Weyl coefficients of SO(p), p = 3, 4, ..., n. The representation functions of ISO(n) are then obtained from those of SO(n,1) or SO(n+1) by contraction. They are expressible as sums over a confluent hypergeometric function with argument 2iγξ, multiplied by Weyl coefficients of SO(p), p = 3, 4, ..., n. This provides an interesting alternate form for the representation functions of ISO(n) obtained previously by Wong and Yeh as a sum over Bessel functions.

Tensor products of positive energy representations of S̃O(3,2) and S̃O(4,2)
View Description Hide DescriptionThe Clebsch–Gordan series of the tensor product of a wide class of unitary irreducible positive energy representations of the universal covering groups of SO(3,2) and SO(4,2) are calculated by comparing weight diagrams.

Sturm–Liouville eigenproblems with an interior pole
View Description Hide DescriptionThe eigenvalues and eigenfunctions of self‐adjoint Sturm–Liouville problems with a simple pole on the i n t e r i o r of the interval [A, B] are investigated. Three general theorems are proved and it is shown that as n→∞, the eigenfunctions more and more closely resemble those of an ordinary Sturm–Liouville problem and λ_{ n } ∼−m ^{2}π^{2}/(B−A)^{2}, just as if there were no singularity. The low‐order modes, however, differ drastically from those of a nonsingular eigenproblem in that (i) both eigenvalues and eigenfunctions are c o m p l e x (despite the fact the problem is self‐adjoint), (ii) the real and imaginary parts of the nth eigenfunction may both have ever‐increasing numbers of interior zeros as B→∞, instead of just (n−1) zeros, and (iii) as B→∞, the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. These results are general, but in order to present quantitative solutions for the low‐order modes, too, special attention is given to the particular case u ^{″}+(1/x−λ)u = 0, (1) with u(A) = u(B) = 0 where λ is the eigenvalue and A and B are of o p p o s i t e signs. For small n, one can obtain the approximation λ_{ n }∼exp[(1+3^{1/2} i)d _{ n }/(2B ^{1/3})]/B, (2) where d _{ n } is the nth root of the Airy function Ai(−z). The imaginary part of (2) shows explicitly how profoundly the interior pole has modified the structure of the eigenproblem. The WKB method, which was used to derive (2), is shown to be accurate for all n. The WKB analysis is of some interest in and of itself. Although the number of WKB ’’transition’’ points is the same as for the half‐century old quantum harmonic oscillator (two), the substitution of the interior pole for one of the turning points has a profound (and fascinating) impact on both the WKB formalism and the numerical results. Thus, although this problem was motivated by the physics of hydrodynamic waves, it is also an extension to both classical Sturm–Liouville theory and to the WKB treatment of eigenvalue problems.

Convergence of Berryman’s iterative method for some Emden–Fowler equations
View Description Hide DescriptionThe equation y′′+λa(x)y ^{α}(x) = 0, 0<x<1, y(0) = y(1) = 0, α≳0, arises in the study of nonlinear diffusion equations connected with crossfield diffusion in plasmas. We show that for a particular choice of starting iterate, the computational method which Berryman developed for this equation does converge, and furnishes upper and lower bounds for y(x).

A direct study of a Marchenko fundamental equation with centripetal potential
View Description Hide DescriptionA direct study of a class of singular (l≠0) fundamental equations is shown to be possible. The method used for this proof follows Marchenko’s nonsingular (l = 0) approach, step by step. Throughout the paper the interest of a simultaneous study of the ’’Marchenko associated equation’’ is stressed. Also it will be shown why the Marchenko approach does not extend to complex interactions. In order to treat a complex interaction one must therefore resort to a new set of ideas.

Chain of the Bäcklund transformation for the KdV equation
View Description Hide DescriptionWe study the chain of the Bäcklund transformation (≡BT) by the example of the KdV equation. The previously obtained chain of the BT, KdV→modified KdV (≡mKdV)→second mKdV, has been extended one step further to the third mKdV case. From this lowest order example, the structure of ’’infinity’’ of the chain process has been foreseen.

A limit on the variation of bounded positive operators
View Description Hide DescriptionAn upper limit is obtained for the rate at which the expectation value, of a bounded positive operator in an arbitrary state, can change with the parameters of a unitary transformation of the state.

Adjoints of nondensely defined Hilbert space operators
View Description Hide DescriptionTo each linear operator T acting in a Hilbert spaceH, an adjoint operator T* is assigned which coincides with the usual adjoint whenever the domain of T is dense in H. General properties of T* are: If H is countably infinite‐dimensional, then the set of all closed operators equals the set of all adjoints; if the domain (range) of T is closed, then so is the domain (range) of T*; T is closable (bounded) if and only if T* is densely (everywhere) defined. Noteworthy corollaries are the closed graph and the closed range theorems, as well as basic crosslinks between adjoints and inverses. Applications to the problem of extending formally self‐adjoint operators are given.

The criticality problem for an exponential atmosphere
View Description Hide DescriptionThe criticality problem for a half‐space with an exponential single scatter albedo is analyzed. Analytic results are presented in the limits of very weak and very strong exponential behavior, and numerical results are given for general exponential behavior.

Conservation laws and discrete symmetries in classical mechanics
View Description Hide DescriptionA method is given for deriving conserved quantities from discrete symmetries in classical mechanics.

On reduction of the four‐dimensional harmonic oscillator
View Description Hide DescriptionThis paper deals with reduction of the four‐dimensional harmonic oscillator by use of a one‐parameter subgroup U(1) of the symmetry group SU(4), U(1) being the symmetry subgroup generated by an ’’angular momentum.’’ The angular momentum determines in the energy surface S ^{7} an ’’energy‐momentum’’ manifoldS ^{3}×S ^{3} on which a subgroup SU(2)×SU(2) of SU(4) acts. The reduction process yields a manifoldS ^{3}×S ^{2} = S ^{3}×S ^{3}/U(1) on which SO(4) acts effectively.

On a ‘‘conformal’’ Kepler problem and its reduction
View Description Hide DescriptionA ’’conformal’’ Kepler problem is defined in order to associate the Kepler problem with the harmonic oscillator. The four‐dimensional conformal Kepler problem which shares an energy surface with the four‐dimensional harmonic oscillator reduces to the ordinary three‐dimensional Kepler problem. By use of the reduction the symmetry group SO(4) of the Kepler problem is brought out from a symmetry subgroup SU(2)×SU(2) of the conformal Kepler problem; the subgroup is the same as a subgroup of the symmetry group SU(4) of the harmonic oscillator.

Classification of orbits of Fokker’s time‐asymmetric relativistic two‐body problem
View Description Hide DescriptionThe requirements that the proper velocities of the particles in Fokker’s time‐asymmetric relativistic two‐body problem be timelike and future pointing restrict the variables ρ_{1} and ρ_{2} associated with the distances between the particles as measured in their rest frames. Employing these restrictions in an algebraic equation relating ρ_{1} and ρ_{2} to the total angular momentum and the total four‐momentum, assumed timelike and future pointing, classifies the physical orbits of the system. The results include orbits similar to those of the nonrelativistic Kepler problem and several new types in which the angular velocity is opposite to the total angular momentum. This information is required for the integration of the equations of motion to determine the orbits in four‐space.

Inverse scattering inverse source theory
View Description Hide DescriptionThe inverse scattering inverse source problem associated with the inhomogeneous Helmholtz wave equation, the (special case) Sturm–Liouville (acoustic wave)equation, and the time‐independent Schrödinger equation is treated . To this end, the concepts of a reference wavevelocity and an associated free reference space Green’s function spectrum are introduced. A modified Kirchhoff surface integral, containing only the gradient of the real part of this free reference space Green’s function spectrum and the fields on a measurement surface is formulated, yielding an integral equation for the unknown fields and sources in the interior of the closed piecewise smooth surface on which the (remotely sensed) fields are known. A well‐posed, analytic closed form solution of this integral equation for the unknown fields and their Laplacians is obtained with the aid of a (modified) spatial Fourier transform in which the reference velocity is continually varied in such a fashion that the Ewald sphere shell sweeps to fill the entire transform space. The unknown potential or medium properties and the unknown sources are then determined algebraically for the inverse scattering and inverse source problems, respectively. The effects of finite sampling density and incomplete observation domain are discussed briefly.