Volume 21, Issue 5, May 1980
Index of content:

Bilateral classes
View Description Hide DescriptionWe introduce ’’bilateral classes’’, a new concept for classifying the elements of groups. Bilateral classes are orbits in a group G under the action of any given subgroup of the direct product G×G. The classification concept presented here encompasses conjugacy classes, cosets, double cosets, and Ree’s σ‐classes as particular cases. It has an interpretation as a classification scheme for bijections between G‐sets under the aspect of symmetry equivalence due to symmetries in both the domain and the range. While double cosets and conjugacy classes correspond to the case of no or complete correlation between operations of the two symmetry groups, our concept also covers the general case of partial correlation. The scope of generalization corresponds to applications in physics. Expressions for the number of bilateral classes are given.

Commutation properties of 2‐parameter groups of isometries
View Description Hide DescriptionWe prove a stronger version of a result due to B. Carter: A cyclic subaction must commute with any 1‐parameter subgroup of a 2‐parameter group of isometries.

On the irreducible representations of groups containing a subgroup of finite index
View Description Hide DescriptionThe objects under consideration are: A group G containing a subgroup S of finite index p, an irreducible representation (= multiplier representation by unitary or by unitary and antiunitary operators on a Hilbert space of arbitrary dimension) U of G, and an irreducible representation W of S. It is shown (1) that the representations U‖S (the restriction of U to S) and W↑G (the representation induced by W) are both orthogonal sums of finitely many irreducible subrepresentations, the number of which does not exceed p; (2) that the multiplicity of W in U‖S equals the multiplicity of U in W↑G if W and U are unitary representations and that these multiplicities are related in a slightly different manner for partially antiunitary representations. For the special case that S is an invariant subgroup, it is shown how the irreducible representations of G can be constructed if the irreducible representations of S and those of certain finite groups are known.

Clebsch–Gordan coefficients for corepresentations. I⊗I
View Description Hide DescriptionA general method is given for determining Clebsch–Gordan coefficients for corepresentations in terms of convenient Clebsch–Gordan coefficients for the normal subgroup, at which the considered Kronecker products are composed of corepresentations of type I only.

Clebsch–Gordan coefficients for corepresentations. I⊗II
View Description Hide DescriptionA general method is given to determine quite generally Clebsch–Gordan coefficients for corepresentations in terms of such ones of the normal subgroup, where the considered Kronecker products are composed of corepresentations of type I and II.

Clebsch–Gordan coefficients for corepresentations. I⊗III
View Description Hide DescriptionA general method is applied to compute Clebsch–Gordan coefficients for corepresentations in terms of such coefficients for the normal subgroup. The considered Kronecker products are composed of corepresentations of type I and III.

Clebsch–Gordan coefficients for corepresentations. II⊗II
View Description Hide DescriptionA general procedure is used to determine quite general Clebsch–Gordan coefficients for corepresentations in terms of convenient Clebsch–Gordan coefficients for the normal subgroup. The considered Kronecker products are composed of corepresentations of type II only.

Clebsch–Gordan coefficients for corepresentations. II⊗III
View Description Hide DescriptionClebsch–Gordan coefficients for corepresentations are determined quite generally in terms of such coefficients for the normal subgroup, at which the Kronecker products are composed of corepresentations of type II and III.

Clebsch–Gordan coefficients for corepresentations. III⊗III
View Description Hide DescriptionBy means of a general method, Clebsch–Gordan coefficients for corepresentations are traced back by simple unitary transformations to convenient Clebsch–Gordan coefficients for the normal subgroup. The considered Kronecker products are composed of corepresentations of type III only.

A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II
View Description Hide DescriptionIt is known through the inverse scatteringtransform that certain nonlinear differential equations can be solved via linear integral equations. Here it is demonstrated ’’directly,’’ i.e., without the Jost‐function formalism that the solution of the linear integral equation actually solves the nonlinear differential equation. In particular, this extends the scope of inverse scattering methods to ordinary differential equations which are found to be of Painlevé type. Some global properties of these nonlinear ODE’s are obtained rather easily by this approach.

A commutator representation of Painlevé equations
View Description Hide DescriptionThis paper further develops the connection between partial differential equations solvable by inverse scattering methods and ordinary differential equations of Painlevé type. The main result given here is that Painlevé equations have a concise algebraic formulation [L,B]=L when written in the Lax representation.

On the remarkable nonlinear diffusion equation (∂/∂x)[a (u+b)^{−} ^{2}(∂u/∂x)]−(∂u/∂t)=0
View Description Hide DescriptionWe study the invariance properties (in the sense of Lie–Bäcklund groups) of the nonlinear diffusionequation (∂/∂x)[C (u)(∂u/∂x)]−(∂u/∂t) =0. We show that an infinite number of one‐parameter Lie–Bäcklund groups are admitted if and only if the conductivity C (u) =a (u+b)^{−} ^{2}. In this special case a one‐to‐one transformation maps such an equation into the linear diffusionequation with constant conductivity, (∂^{2}ū/∂x̄^{2})−(∂ū/∂t̄) =0. We show some interesting properties of this mapping for the solution of boundary value problems.

Towards a factorization of M _{4}
View Description Hide DescriptionIt may be desirable to eliminate M _{4} as the underlying manifold in which some physical theories are cast, and recast these theories in the space ’’√M _{4}.’’ We investigate some of the properties of one such space, which we denote by S _{8}. S _{8} can be coordinatized by real eightcomponent spinors. The spinor algebra on this space is developed in this paper. It is shown that a (nondegenerate) spinor in S _{8} determines an orthogonal tetrad on M _{4} (the set of these spinors determines the space of orthogonal frames over M _{4}), and that this spinor corresponds to a ’’particle.’’ A simple geometrical interpretation of the Dirac equation arises in arriving at this correspondence.

On choosing the points in product integration
View Description Hide DescriptionA product‐integration rule for the integral F^{ b } _{ a } k (t) f (t) d t is a rule of the form J^{ n } _{ i=1} w _{ i } f (t _{ i }), with the weights w _{1},...,w _{ n } chosen so that the rule is exact if f is any linear combination of a chosen set of functions φ_{1},...,φ_{ n } . For some choices of {φ_{ j }}, including the polynomial case, the points {t _{ i }} need to be carefully chosen if reliable results are to be obtained. In this paper known convergence results for the polynomial case with well‐chosen points are summarized and illustrated, and extended to some nonpolynomial cases, including one proposed by Y.E. Kim for use in solving the three‐body Faddeev equations. The convergence theorems yield practical prescriptions for choosing the points {t _{ i }} .

Decaying states in the rigged Hilbert space formulation of quantum mechanics
View Description Hide DescriptionWithin the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self‐adjoint Hamiltonian with complex eigenvalue and a Breit–Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.

Resonances in the Klein–Gordon theory of the relativistic Stark effect
View Description Hide DescriptionExistence of resonances is proved for the time‐independent Klein–Gordon equation describing the interaction of a charged particle with an external uniform field of small strength F in addition to the Coulomb attraction. It is further shown that the resonances reduce to the exactly known bound states of the problem as F→0, and to the resonances of the nonrelativistic Stark effect as c→∞.

Continuum calculus. IV. The Laplace transform method in the evaluation of the Feynman path integrals with a Gaussian measure and applications in quantum mechanics
View Description Hide DescriptionThe continuum calculus, proposed earlier [J. Math. Phys. 17, 1988 (1976)], is here applied to the development of a functional version of Laplace transform and a method for the evaluation of path integrals containing a Gaussian‐like measure. Two methods in functional integration are proposed. First, the Gaussian integral for polynomial functionals and consequently functionals that can be expanded in Taylor series are examined. Formula (2.12) is derived. Next, we define the Laplace transform in the function space through the weak distribution formulation of Skorohod. Comparison of both approaches enables us to determine the expression of the functional integeral through a series of Laplace transformations. The second formula is given in Eq. (4.5). The latter formula is applicable to all Laplace transformable functionals. As an illustration of the utility of the formulas derived, we evaluate the integral of a cosine functional by methods 1 and 2, and obtain consistent results. Further applications to quantum mechanics are also presented. We examine the cases of a free particle,the quantum harmonic oscillator, the forced oscillator, and charged particle in a magnetic field. In all cases, we obtain correct results in comparison with known expressions. A numerical procedure is employed in the calculation of infinite products. The usefulness of the p‐integral method is stressed.

Systematic improvement of Hall–Post–Stenschke lower bounds to eigenvalues in the few‐body problem
View Description Hide DescriptionA method for systematically improving Hall–Post–Stenschke lower bounds to the bound stateeigenvalues of three‐body Schrödinger equations is given. The improved bounds are obtained by solving coupled one variable integral equations; the bounds get better as the number of coupled equations is increased. The method generates explicit wave functions which can be used to obtain complementary upper bounds via the Rayleigh–Ritz variational method. Either identical or nonidentical particles can be handled. The method is illustrated by calculations for three identical particles bound by Hooke’s law forces. A brief discussion of extensions to more than three particles is given.

An exactly solvable one‐dimensional three‐body problem with hard cores
View Description Hide DescriptionThree identical particles in one dimension interact via a potential which is infinite whenever one or more of the interparticle separations is less than a or greater than b, and zero when all interparticle separations lie between a and b. Their Schrödinger equation is solved by reducing it to the exactly solvable problem of the two dimensional Helmholtz equation inside an equilateral triangle.

Test of a method for finding lower bounds to eigenvalues of the three‐body problem
View Description Hide DescriptionThe simplest version of a method for systematically improving Hall–Post–Stenschke (HPS) lower bounds to eigenvalues is tested on an exactly soluble one dimensional three‐body problem with hard cores. Significant improvement over the HPS bound is obtained, but considerable room for additional improvement remains.