Volume 23, Issue 6, June 1982
Index of content:

An upper bound on the number of eigenvalues of an infinite dimensional Jacobi matrix
View Description Hide DescriptionAn upper bound is given on the number of eigenvalues of a class of infinite dimensional Jacobi matrices. The theorem presented is a discrete analog of the celebrated result of V. Bargmann on the number of bound states of the Schrödinger equation.

Permutational properties of the generalized Clebsch–Gordan coefficients
View Description Hide DescriptionThe properties of the permutational symmetry of the 3Γγ symbols for an arbitrary compact group have been generalized to a multiple Kronecker product of irreducible representations. It has been shown that these properties of the corresponding NΓγ symbols under a permutation of their columns are related to a choice of the coupling schemes for the appropriate polyads Γ_{1}⋅⋅⋅Γ_{ N }. The simplest permutational symmetry, i.e., associated with a Young diagram for the symmetric group Σ_{ N }, is, in general, incompatible with schemes having definite intermediate representations and, in the case of mixed symmetry, does not preserve the absolute value of a NΓγ symbol. The case of N = 4 for SU(2) has been studied in detail.

A new approach to permutation group representation
View Description Hide DescriptionThe new approach to group representation theory is applied to the treatment of the permutation group. It is shown that obtaining (1) the primary characters and the fractional parentage coefficients, (2) the Yamanouchi bases and the Clebsch–Gordan coefficients, and (3) the irreducible matrix elements of the permutation group S ( f), are all simplified to a unified procedure — diagonalizing a certain operator in the corresponding representation. The operator to be diagonalized for the above three problems is (1) the 2‐cycle class operator C _{ f } of S ( f), (2) an appropriate linear combination of the f‐1 2‐cycle class operators C _{ f }, C _{ f−1},...,C _{2} of the group chain S( f)⊇ S( f−1)⊇...⊇ S(2), and (3) an appropriate linear combination of 2f−3 2‐cycle class operators C _{ f }, C _{ f−1},...,C _{2}, C_{ f−1},...,C_{2}, C_{ i } being the 2‐cycle class operator of the subgroup S(i) of the state permutation group S(f), respectively. This method, the eigenfunction method, is simpler in concept, yet more powerful in practical calculations.

Structure and representations of the symmetry group of the four‐dimensional cube
View Description Hide DescriptionIn this paper we give, explicitly, the description of the structure, the characters, and the complete system of irreducible representations of the hyperoctahedral group in four dimensions, which we call W_{4} (from the German ’’Würfel’’). In a second step, we do the same for the subgroup SW_{4}, which is formed by all pure rotations contained in W_{4}.

Infinite‐dimensional representations of the graded Lie algebra (Sp(4):4). Representation of the para‐Bose operators with real order of quantization
View Description Hide DescriptionInfinite‐dimensional representations for the system of the two para‐Bose operators, which generate the graded Lie algebra (GLA) (Sp(4):4), are constructed by using the irreducible representations (d i s c r e t e s e r i e s) of the Lie algebra Sp(4)≊SO(3,2). It is shown that there are four kinds of the irreducible representations of the GLA’s (Sp(4):4), i.e., cases (I), (II), (III), and (IV). Case (I) is described by the three irreducible representations of Sp(4) and corresponds to the para‐Bose quantization with real order of the quantization greater than 1 and case (IV) corresponding to the ordinary Bose quantization by the two irreducible representations of Sp(4). Cases (II) and (III), which are described by the four and two irreducible representations of Sp(4), respectively, and cannot be obtained by the method of Fock space, express the representations of the graded Lie algebra (Sp(4):4).

A family of sums of products of Legendre functions
View Description Hide DescriptionWe present simple analytic expressions for a few sums of products of Legendre functions, of the type J^{∞} _{ n = 0}(2n+1)P^{α} _{ n }(x)P^{β} _{ n }(y) P^{γ} _{ n }(z)Q^{μ} _{ n }(n).

Variational calculation of the multipole potential from an arbitrary localized charge distribution
View Description Hide DescriptionVery recently Painter has developed a method for solving Poisson’s equation as a set of finite‐difference equations for an arbitrary localized charge distribution ρ(r) that is expanded in a partial‐wave representation as ρ(r) = J_{ L }ρ_{ L }(r)Y _{ L }(r̂), where L denotes l and m. In the present work a variational principle is established, and a possible approach is outlined, for obtaining approximate partial‐wave coefficients V _{ L }(r) of the potential V(r) = J_{ L } V _{ L }(r)Y _{ L }(?).

A nonstandard infinite dimensional vector space approach to Gaussian functional measures
View Description Hide DescriptionNonstandard analysis is used to apply the usual concepts of finite dimensional vector spaces in order to define various Gaussian functional measures without using a limiting process. The approach is to define matrices on the infinite dimensional space and use straightforward techniques to determine the properties of the functions the measures are concentrated on. The power of nonstandard analysis allows one to work directly with infinite and infinitesimal quantities and ’’visualize’’ certain sets upon which the basic Gaussian form is or is not infinitesimal. The relation of the choice of Gaussian to the function (process) properties remains heuristic since a proof of the Holder continuity of the Weiner paths is not complete, but remains at a local (infinitesimal) level. However, given the Holder continuity the nonbounded variation property easily follows. Higher derivative Gaussian measures are also easily developed and their analytic properties displayed along with their covariances. By using Fourier analysis on finite abelian groups transferred to the nonstandard universe and applied to hyperfinite abelian groups a rigorous transformation from the discrete form of the Weiner measure to a Fourier series form is accomplished. It is shown here that the functions (processes) are infinitesimally close to those of the discrete version of the measure. The Fourier approach is also extended to more general measures. Finally, some speculations show directions in which this direct approach to Gaussian functional measures can be extended and generalized.

Asymptotic behavior of the nonlinear diffusion equation n _{ t } = (n ^{−1} n _{ x })_{ x }
View Description Hide DescriptionThe asymptotic behavior of the equation n _{ t } = (ln n)_{ x x } is studied on the finite interval 0⩽x⩽1 with the boundary conditionsn(0,t) = n(1,t) = n _{0} and initial data n(x,0)⩽n _{0}. We prove that asymptotically ln[n(x,t)/n _{0}]→A exp(−π^{2} t/n _{0})2^{1/2} sin πx and also provide rigorous upper and lower bounds on the asymptotic amplitude A in terms of integrals of nonlinear functions of the initial data. The rigorous bounds are compared to values of A obtained from computer experiments. The lower bound L = (2^{3/2}/π)exp[li(1+Q)−γ], where li is the logarithmic integral, γ is Euler’s constant, and Q = (π/2)F[n(x,0)/n _{0}−1]sin πx d x, is found to be the best known estimate of A.

New Stokes’ line in WKB theory
View Description Hide DescriptionThe WKB theory for differential equations of arbitrary order or integral equations in one dimension is investigated. The rules previously stated for the construction of Stokes’ lines for Nth‐order differential equations,N⩾3, or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path. This paradox is resolved by the demonstration that new Stokes’ lines can arise when previously defined Stokes’ lines cross. A new formulation of the WKB problem is given to justify the new Stokes’ lines. With the new Stokes’ lines, the asymptotic forms can be shown to be independent of path. In addition, the WKB eigenvalue problem is formulated, and the global dispersion relation is shown to be a functional of loop integrals of the action.

Some more developments on operators in Krein space. The exponential map
View Description Hide DescriptionThis paper generalizes the work of J. L. B. Cooper on symmetric operators in a Hilbert space to Pontrjagin and Krein spaces. Existence (and uniqueness) of a solution for Schrödinger’s equationdψ(t)/d t = i Aψ(t), ψ(0) = φ∈Π [ = Π^{+}(+̇)π̄] for the bounded decomposable symmetric operator A (the self‐adjoint operator A) is studied. Also, the existence of a solution for (1/i)∂ψ(t)/∂t = A*ψ(t), ψ(0) = φ∈Π, where A is a cross‐bounded symmetric operator in Π, is discussed. Finally, existence and uniqueness of the equation (1/i)(∂ψ(t)/∂t) = Aψ(t), ψ(0) = φ∈Π_{ k }, where A is a self‐adjoint operator in the Pontrjagin space Π_{ k }, is studied, and the fact that the maps φ→ψ(t) form a group of unitary operators on Π_{ k } is deduced.

A new method for the asymptotic evaluation of a class of path integrals
View Description Hide DescriptionA general method of calculating the asymptotic behavior of a class of Wiener path integrals is given. These integrals are averages of functionals of the ’’local time.’’ The technique is essentially a variation on the well‐known ’’replica’’ method now widely used in condensed matter physics, combined with the Laplace method for evaluating integrals containing a large parameter. The leading term is given, and from the construction one sees that the error is typically of order 1/t.

Stochastic optimal control and quantum mechanics
View Description Hide DescriptionClassical mechanics is formulated as a kind of deterministic optimal control. The simplest, from the optimal control point of view, stochastic generalization of classical mechanisms is submitted. The connections between stochastic mechanics and quantum mechanics are shown, thanks to the stochastic optimal control method.

On the dynamics of diffusions and the related general electromagnetic potentials
View Description Hide DescriptionIn this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one‐to‐one correspondence between solutions of certain Schrödinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrödinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.

Properties of three‐dimensional Cartesian tensors. I. Some properties of irreducible tensors
View Description Hide DescriptionIt is shown that the decomposition of three‐dimensional Cartesian tensors into their parts which are irreducible under the rotation group is made conceptually simple by explicitly expressing these parts in an embedded form.

Properties of three‐dimensional Cartesian tensors. II. Arbitrarily oriented tensors expressed in generalized spherical functions. Ranks through 4
View Description Hide DescriptionExplicit expressions for the rotation of Cartesian tensors of ranks through 4 are obtained. These are given in terms of the irreducible components and the generalized spherical functions. They can be obtained in terms of the reducible components by the use of results which were obtained previously. As a simple illustration of an application of the results, an expression is obtained for the energy of an octupole as a function of the orientation of the octupole in an inhomogeneous field.

Properties of three‐dimensional Cartesian tensors. III. Concerning tensor averages
View Description Hide DescriptionResults from the two earlier papers in this series are used to investigate some average tensorial properties of systems in terms of the tensorial properties of their subsystems. Special attention is given to the Boltzmann distribution in which the orientation energy of a subsystem is assumed to be expressible as a power series in the components of a single vector which is fixed in the laboratory system. The orientation probability density is obtained in general form for subsystems with no symmetry (point group C _{1}), and they are given in more explicit form for a few cases of higher symmetry. General expressions for tensor averages are given, and they are applied to a special case as an example.

Construction of new integrable Hamiltonians in two degrees of freedom
View Description Hide DescriptionA new procedure for deriving integrable Hamiltonians and their constants of the motion is introduced. We term this procedure the truncation program. Integrable Hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter ε. The relationship between this program and the Whittaker program in two degrees of freedom is discussed. Integrable Hamiltonians occurring in the Whittaker program (a generalization of Whittaker’s work) possess constants of the motion which are polynomials in the momentum coordinates. Many previously known integrable Hamiltonians are derived. A new family of integrable double resonance Hamiltonians and a new family of integrable Hamiltonians of the form (p ^{2} _{1}+p ^{2} _{2})/2+V(q _{1}, q _{2}) are derived.

Dyadic Green functions for the time‐dependent wave equation
View Description Hide DescriptionThe theory of dyadic Green functions for a transient electromagnetic field, which obeys the vector wave equation, is presented within the framework of the theory of distributions. First, the the elementary solution of the scalar wave equation is derived, and then it is used to find the general solution of that equation. After establishing the equivalence between Maxwell’sequations and the time‐dependent vector wave equation, the dyadic elementary solution is derived and applied to solve the equation. Further properties of dyadic Green functions for the wave equation are derived within the heuristic approach to the theory of Green functions. The paper includes a collection of formulas from the theory of distributions intended to help readers who are not familiar with the subject.

Single integral equation for wave scattering
View Description Hide DescriptionWhen a wave interacts with an obstacle, the scattered and transmitted fields can be found by solving a system of integral equations for two unknown fields defined on the surface of the body. By choosing a more appropriate unknown function, the system of equations is reduced to a single singular integral equation of the first kind. This reduction is done here for transient and monochromatic waves, for a scalar field that obeys the wave equation, and for electromagnetic fields that obey Maxwell’sequations.