Volume 24, Issue 5, May 1983
Index of content:

SU(2)×SU(2) shift operators and representations of SO(5)
View Description Hide DescriptionSU(2)×SU(2) shift operators analogous to the SU(2) shift operators developed and used by the author for the classification and analysis of representations of Lie algebras in an SU(2) or SO(3) basis are obtained for the SU(2)×SU(2) Lie algebra in the case where one has an additional set of operators forming an irreducible four‐dimensional tensor representation of SU(2)×SU(2). The shift operators obtained are used to treat the representations of SO(5) in an SU(2)×SU(2) basis.

On the relations between irreducible representations of the hyperoctahedral group and O(4) and SO(4)
View Description Hide DescriptionIn this paper we describe the relations between the irreducible representations of the hyperoctahedral group in four dimensions and irreducible, low‐dimensional representations of the orthogonal groups O(4) and SO(4).

Nonscalar extension of shift operator techniques for SU(3) in an O(3) basis. III. Shift operators of second degree in the tensor components
View Description Hide DescriptionShift operators Q ^{ k } _{ l } (−2≤k≤2) of second degree in the tensor components q _{μ} (−2≤μ≤2) are constructed. Relations connecting quadratic shift operator products of the type O ^{ j } _{ l+k } Q ^{ k } _{ l } or Q ^{ k } _{ l+j } O ^{ j } _{ l } , and of the type Q ^{ j } _{ l+k } Q ^{ k } _{ l } are derived. The usefulness of these relations is demonstrated by the example of the O ^{ O } _{ l } ‐ and Q ^{ O } _{ l } ‐eigenvalue calculation for various irreducible respresentations (p,q) of SU(3).

Indecomposable representations for para‐Bose algebra
View Description Hide DescriptionA general study of the representations of the graded Lie algebra of para‐Bose oscillators is given. Besides realizing the standard representations, we also find some interesting indecomposable (not fully reducible) representations.

Harmonic polynomials invariant under a finite subgroup of O(n)
View Description Hide DescriptionIn this paper, an algorithm is described which allows a systematic computation of harmonic polynomials of a given degree invariant under a finite subgroup of the group O(n). An application of the algorithm to the octahedral (cubic) subgroup is given.

A Galerkin method and nonlinear oscillations and waves
View Description Hide DescriptionA Galerkin method is developed as a generalization of the variational averaging method to deal with problems with dissipation. Some nonlinear oscillations, nonlinear waves, and nonlinear stability problems are studied to illustrate the application of the new method. It is demonstrated that when the dissipative parameter is small, the solutions agree with those obtained by other established methods.

The general theory of R‐separation for Helmholtz equations
View Description Hide DescriptionWe develop the theory of R‐separation for the Helmholtz equation on a pseudo‐Riemannian manifold (including the possibility of null coordinates) and show that it, and not ordinary variable separation, is the natural analogy of additive separation for the Hamilton–Jacobi equation. We provide a coordinate‐free characterization of variable separation in terms of commuting symmetry operators.

Some infinite series of products of Legendre and gamma functions
View Description Hide DescriptionWe derive closed expressions for some infinite series of products of Legendre functions and gamma functions. A particular series has been used to obtain the partial‐wave projected quantum mechanical Coulomb transition matrix in closed analytic form for all partial waves,l=0, 1,⋅⋅⋅.

A simple proof of a transformation formula for elliptic integrals
View Description Hide DescriptionA very simple proof of a quadratic transformation formula for elliptic integrals found by Carlson in 1977 is given.

Expression for y _{ L M }[(r _{1}Λr _{2})Λr _{3}]
View Description Hide DescriptionThe solid harmonic y _{ L M }[(r _{1}Λr _{2})Λr _{3}] is expressed in terms of the spherical harmonics Y _{ L 1 M 1 }(r̂_{1}) , Y _{ L 2 M 2 }(r̂_{2}), and Y _{ L 3 M 3 }(r̂_{3}). The calculation of the coefficients in the given expansion in terms of 9−j symbols explicitly justifies the form given by y _{ L M }[(r _{1}Λr _{2})Λr _{3}] J_{ L 1 L 2 L 12 L 3 L M 1 M 2 } A(L _{1},L _{2},L _{12},L _{3},L) ×〈L _{1},M _{1};L _{2},M _{2}‖L _{1},L _{2},L _{12} M _{12}〉 ×〈L _{12},M _{12};L _{3},M _{3}‖L _{12} L _{3} L M〉 ×Y _{ L 1 M 1 }(r̂_{1})Y _{ L 2 M 1 }(r̂_{2}) ×Y _{ L 3 M 3 }(r̂_{3}).

Superposition principles for matrix Riccati equations
View Description Hide DescriptionA superposition rule is obtained for the matrix Riccati equation (MRE) Ẇ=A+W B+C W+W D W [where W(t), A(t), B(t), C(t), and D(t) are real n×n matrices], expressing the general solution in terms of five known solutions for all n≥2. The symplectic MRE (W=W ^{ T }, A=A ^{ T }, D=D ^{ T }, C=B ^{ T }) is treated separately, and a superposition rule is derived involving only four known solutions. For the ‘‘unitary’’ and GL(n,R) subcases (with D=A and C=B ^{ T }, or D=−A and C=B ^{ T }, respectively), superposition rules are obtained involving only two solutions. The derivation of these results is based upon an interpretation of the MRE in terms of the action of the groups SL(2n,R), SP(2n,R), U(n), and GL(n,R) on the Grassman manifold G_{ n }(R^{2n }).

Restricted multiple three‐wave interactions: Integrable cases of this system and other related systems
View Description Hide DescriptionRestricted multiple three‐wave interactions, in which a set of wave triads interact through one shared wave, are discussed. It is shown that this system is integrable when all triads have equal coupling coefficients regardless of the frequency mismatches. This system is then used as a starting point from which to determine integrable cases of a more general class of three‐wave interactions.

Manifestly covariant, coordinate‐free dyadic expression for planar homogeneous Lorentz transformations
View Description Hide DescriptionParametrizing a planar homogeneous Lorentz tranformation P by any timelike or spacelike vector b lying in the transformation plane and its transform a≡P b yields a dyadic expression for P with several advantages: It provides an immediate solution to the problem of finding a homogeneous Lorentz transformation converting a given timelike or spacelike vector into a second similar vector. Its manifestly covariant, coordinate‐free form is valid in any Lorentz frame and reduces easily to coordinate form. It unifies timelike (including boosts), spacelike (including pure spatial rotations), and null planar transformations and also orthochronous and nonorthochronous planar transformations into a single form; these classifications depend on the vectors a and b. Only if a=−b does the expression fail, but then its limit as a→−b still exists and provides a valid expression for P.

Can poles change color?
View Description Hide DescriptionThe definition of the total nonabelian charge (‘‘color’’) in a classical Yang–Mills theory is shown to require a careful analysis of the boundary conditions at infinity imposed on the potentials and on gauge transformations. The color current of a nonabelian plane wave is found to be different from zero in the transverse gauge, though it vanishes in the null gauge. The color charge of a single pole, described by the Liénard–Wiechert potentials, is constant by virtue of the Yang–Mills equations. An approximate computation indicates that the total color charge of a system of particles may change in time, as a result of radiation. To make this result meaningful, it is necessary to find a method of fixing the allowed gauge transformations to those having a direction‐independent limit at infinity.

Time evolution kernels: uniform asymptotic expansions
View Description Hide DescriptionFor a wide class of self‐adjoint Schrödinger Hamiltonians, a detailed description of the time evolution kernel is obtained. In a setting of a d‐dimensional Euclidean space without boundaries, the Schrödinger Hamiltonian H is the sum of the negative Laplacian plus a real‐valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N‐body problem, since they do not necessarily decrease as ‖x‖→∞. An asymptotic expansion in the complex parameter z, around z=0, is derived for the family of kernels U _{ z }(x,y) corresponding to the analytic semigroup {e ^{−z H }:Re z>0}, which is uniform in the coordinate variables x and y. The asymptotic expansion has a simple semiclassical interpretation. Furthermore, an explicit bound for the remainder term in the asymptotic expansion is found. The expansion and the remainder term bound continue to the time axis boundary z=i t/ℏ (t≠0) of the analytic semigroup domain.

Time evolution of the Wigner function
View Description Hide DescriptionIn this paper we give a partial answer to the problem: When does an initially non‐negative Wigner function remain non‐negative under the effect of the time evolution? We show that, for pure states, this is the case for linear systems only; to prove this we use the fact that the Wigner function is non‐negative if and only if the wavefunction is Gaussian. We also prove that the Green’s solution of the evolution equation of the Wigner function, which in the framework of probability theory corresponds to the conditional probability density, takes on negative values. We utilize a theorem, about moments, borrowed from Pawula. We conclude that the Wigner phase‐space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation.

Transfer matrices for one‐dimensional potentials
View Description Hide DescriptionThe one‐dimensional Schrödinger equation can be written as a first‐order multicomponent equation by considering ψ and dψ/d x, or combinations thereof, as independent variables. A potential barrier is then represented by a matrix belonging to one of the homomorphic groups SU(1,1), SO(2,1), Sp(2,R), or SL(2,R). The relationship between these groups is clarified. In various applications, one of them may turn out more convenient than others. In particular, SO(2,1), which is obtained by using as a basis some bilinear combinations of ψ and dψ/d x, leads to remarkable results: The Schrödinger wavefunction is represented by a trajectory on a unit hyperboloid; a periodic potential corresponds to a pseudorotation around a fixed axis; a random potential gives a random walk on the hyperboloid. This method can also be used to calculate bound states (in potential wells) and may have many other interesting applications.

A probabilistic formulation of quantum theory. III
View Description Hide DescriptionA representation of the expectation value of quantum mechanics, which recently has been set up for normal states, is generalized to singular states. It is shown that states can be represented by finitely additive measures on the state space, which are σ‐additive if and only if the states are normal.

Nonlinear time‐dependent anharmonic oscillator: Asymptotic behavior and connected invariants
View Description Hide DescriptionThe motion of a particle in a potential decreasing with time as ‖X‖^{ n } is considered. Different time and space rescaling are considered in order to obtain the asymptotic solutions. The validity of adiabatic invariants is discussed. The classical critical case corresponds to the obtainment of self‐similar solutions for the quantum problem.

Complex‐potential description of the damped harmonic oscillator
View Description Hide DescriptionThe multidimensional damped harmonic oscillator is treated by means of a non‐self‐adjoint Hamiltonian with complex potential. The propagator referring to the evolution semigroup is evaluated from the Lie–Trotter formula. The one‐dimensional case is discussed in detail with the following results: (a) the nondamped limit gives the correct propagator including the Maslov phase factor, (b) for some initial conditions, the classical limit of the solution can differ from the behavior of the classical damped oscillator, the difference being negligible in the case of weak damping, and (c) the point spectrum of the considered pseudo‐Hamiltonian is found.