Volume 26, Issue 4, April 1985
Index of content:

Riccati equations and perturbation expansions in quantum mechanics
View Description Hide DescriptionGeneral perturbation expansions, which allow corrections to any order to be written in quadrature, are presented for Riccati and other nonlinear first‐order equations. These results are valid for eigenfunctions which are free of poles and zeros. A Riccati equation suitable for a Schrödinger or Klein–Gordon particle in a central field is expanded for a general state, with corrections to all orders expressed in quadrature. A general Riccati equation for a meromorphic eigenfunction leads to a similar expansion with corrections to all orders, including corrections to the zeros and simple poles, expressed in quadrature. This form is suitable for a Dirac particle in a central field but is more general. The general results are applied to specific examples from the literature.

Closed formula for the product of n Dirac matrices
View Description Hide DescriptionThe product of nDirac γ matrices is evaluated in terms of traces of at most (n+3) γ matrices. This leads to a method that can be used to generalize identities for a correlated product of γ matrices. The expression for the n γ matrix product also provides a method for the evaluation of a scalar, a pseudoscalar, a vector, an axial vector, and a second rank antisymmetric tensor associated with a one‐line fermion amplitude that drastically simplifies the squaring of fermionic amplitudes. The general spin and particle–antiparticle dependence of the squared amplitude is given.

The relation of a theory of countable sets to the field equations of physics
View Description Hide DescriptionThe systematic development of mathematics is based on the theory of sets. We present an axiom system in which, in contradistinction to the usual theories, it seems possible to define formal provability yet some useful mathematics can be derived. Some features of the theory suggest that these axioms can provide a possible new foundation for mathematical physics.

Spinors as fundamental objects
View Description Hide DescriptionWe define, on the algebraic Dirac spinor space Ψ, some operators D ^{±} and T ^{±}. By means of them we show how the fundamental operations of Hermitian conjugation, complex conjugation, bar conjugation, and so on may be introduced in the Clifford algebra approach. These definitions depend on the geometrical property of the pure imaginary unit i of the DiracalgebraD _{ s,t } and are the same only for mod 4 dimensions of vector space‐times R ^{ s,t }. Furthermore, on the set Ψ×Ψ we introduce equivalence relations R _{±} and define bijections χ_{±} between Ψ×Ψ/R _{±} and D _{ s,t }. We investigate some properties of χ_{±} and give the necessary and sufficient conditions for u∈D _{ s,t } to belong to some minimal left ideal of D _{ s,t }. Next we use the decomposition of the DiracalgebraD _{ s,t } onto the Dirac spinor spaces to demonstrate two different ways of an action of any element s∈Spin(x,t). These considerations throw a new light onto the problem of the covariant derivative on the bundle of algebraic spinors over a space‐time manifold.

Infinitesimal symmetry transformations. II. Some one‐dimensional nonlinear systems
View Description Hide DescriptionThe converse problem of similarity analysis is discussed for the infinitesimal symmetry transformations of ordinary second‐order differential equations which are nonlinear in ẋ (and may be linear or nonlinear in x). A natural classification of the problem arises, according to the highest order N of nonlinearity in ẋ. The completely general maximal Lie algebra is obtained for the case N≤3. In the case N≥4 one has, besides the system of differential equations for the infinitesimal generators, an extra set of anholonomic constraints, which operates as a symmetry‐breaking mechanism producing a strong reduction in the number of surviving parameters. Miscellaneous examples are given, which illustrate some features of similarity analysis of nonlinear systems. The infinitesimal point transformation symmetries of the Van der Pol oscillator are also briefly discussed.

Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics
View Description Hide DescriptionDecomposition formulas of general exponential operators in a Banach algebra and in a Lie algebra are presented that yield a basis of Monte Carlo simulation of quantum systems. They are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena. A global approximation method of transient phenomena is also formulated on the basis of decomposition formulas. It is applied to the laser model as a simple example.

N‐dimensional spinors: Their properties in terms of finite groups
View Description Hide DescriptionA classification scheme is presented for the finite multiplicative group generated by the gamma matrices associated with a given Clifford algebra. This group reflects the periodicities observed in n‐dimensional spinors, and its representations and other properties are studied, thus highlighting the dependence on the number of spacelike and timelike vectors. The reality of the representations is examined and tabulated; application is made to the imposition of Majorana and Weyl conditions.

Perturbation of Schrödinger Hamiltonians by measures—Self‐adjointness and lower semiboundedness
View Description Hide DescriptionWe study the Hamiltonians for nonrelativistic quantum mechanics in one dimension, in terms of energy forms ∫‖d f/d x‖^{2} d x+∫‖ f ‖^{2} d( μ −ν), where μ and ν are positive, not necessarily finite measures on the real line. We cover, besides regular potentials, cases of very singular interactions (e.g., a particle interacting with an infinite number of fixed particles by ‘‘delta function potentials’’ of arbitrary strengths). We give conditions for lower semiboundedness and closability of the above energy forms, which are sufficient and, for certain classes of potentials (e.g., μ−ν a signed measure), also necessary. In contrast to the results in other approaches, no regularity conditions and no restrictions on the growth of the measures μ and ν at infinity are needed.

Sums of products of ultraspherical functions
View Description Hide DescriptionAnalytical expressions for the sum ∑^{∞} _{ n=0}(n+λ)[Γ(n+1)Γ(n+2λ)]^{2} ×C ^{λ} _{ n }(x)C ^{λ} _{ n }( y) ×C ^{λ} _{ n }(z)D ^{λ} _{ n }(u) are given, where C ^{λ} _{ n } and D ^{λ} _{ n } are ultraspherical polynomials and functions of the second kind, respectively, on the sets {‖x‖,‖ y‖,‖z‖<1, u>1} and {‖x‖,‖ y‖,‖z‖,‖u‖<1}.

Integrals of three Bessel functions and Legendre functions. I
View Description Hide DescriptionIntegrals of products of three Bessel functions of the form ∫^{∞} _{0} t ^{λ−1} J _{μ}(a t)J _{ν} ×(b t)H ^{(1)} _{ρ}(c t)d t are calculated when some relations exist between the indices λ, μ, ν, ρ: in these cases, the Appell function F _{4} factorizes into two hypergeometric functions of one variable, so that analytical continuation is possible. New results are given, mainly when a, b, and c are real and positive and ‖a−b‖<c<a+b, which correspond to most physical situations.

Integrals of some three Bessel functions and Legendre functions. II
View Description Hide DescriptionIntegrals of three Bessel functions of the form ∫^{∞} _{0} J _{μ}(a t)J _{ν}(b t) ×H ^{(1)} _{ρ}(c t)d t are calculated when μ,ν,ρ,a,b, and c are arbitrary real numbers. For this, use is made of the factorization of the Appell functionF _{4} in two hypergeometric functions. Further simplifications occur if μ=±ν or ρ=±1/2. New results are given, mainly when real a, b, and c satisfy the inequalities ‖a−b‖<c<a+b, which correspond to most physical situations.

Bessel function expansions of Coulomb wave functions
View Description Hide DescriptionFrom the convergence properties of the expansion of the function Φ_{ l }∝r ^{−l−1} F _{ l } in powers of the energy, we successively obtain the expansions of F _{ l } and G _{ l } as single series of modified Bessel functions I _{2l+1+n } and K _{2l+1+n }, respectively, as well as corresponding asymptotic approximations of G _{ l } for ‖η‖→∞. Both repulsive and attractive fields are considered for real and complex energies as well. The expansion of F _{ l } is not new, but its convergence is given a simpler and corrected proof. The simplest form of the asymptotic approximations obtained for G _{ l }, in the case of a repulsive field and for low positive energies, is compared to an expansion obtained by Abramowitz.

Exponential convergence for nonlinear diffusion problems with positive lateral boundary conditions
View Description Hide DescriptionIt is established that the solution u of u _{ t }=Δ(u ^{ m })>0, with positive initial data, positive lateral boundary data, and positive exponent m, converges exponentially to the solution v of the corresponding stationary equation Δ(v ^{ m })=0. The analysis also provides the form of the leading contribution to the difference (u−v).

A simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics
View Description Hide DescriptionIn this article a simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics is presented. Our derivation is based upon properties of the differential operator Y^{ m } _{ l }(∇), which is obtained from the regular solid harmonic Y^{ m } _{ l }(r) by replacing the Cartesian components of r by the Cartesian components of ∇. With the help of this differential operator Y^{ m } _{ l }(∇), which is an irreducible spherical tensor of rank l, the addition theorems of the anisotropic functions are obtained by differentiating the addition theorems of the isotropic functions. The performance of the necessary differentiations is greatly facilitated by a systematic exploitation of the tensorial nature of the differential operator Y^{ m } _{ l }(∇).

On the body of supermanifolds
View Description Hide DescriptionThe problem of constructing the body of a G ^{∞}manifold is considered. It is shown that any such manifold is foliated, and the body is defined to be the space of the leaves of this foliation. Under certain regularity conditions on the foliation, the body is a smooth finite‐dimensional real manifold.

Harmonic analysis on the Euclidean group in three‐space
View Description Hide DescriptionWe develop the extensive harmonic analysis on the universal covering group of the Euclidean group in three‐space.

Asymptotic eigenvalue degeneracy for a class of one‐dimensional Fokker–Planck operators
View Description Hide DescriptionLet f(x), x∈R, be a fourth‐degree polynomial with lim_{‖x‖→∞} f(x) =+∞ with two minima, and let L _{ε}( ⋅ )=ε^{2}/2 ∂^{2}( ⋅ )/∂x ^{2}+(∂/∂x) ×(^{(∂f/∂x }/_{( ⋅ )})) be the corresponding Fokker–Planck operator. We study the spectrum of L _{ε} in the limit ε→0. We show that in the limit ε→0 the spectrum of L _{ε} degenerates in the spectrum of three decoupled harmonic oscillators.

Stochastic difference equations for a spin system
View Description Hide DescriptionAn N‐spin model is given with a discrete‐time evolution specified by a system of stochastic difference equations. A Markov chain associated with the evolution is decomposed into two, nonhomogeneous, absorbing Markov chains. Analysis of each chain yields the probability, given a specific initial state, of ultimate absorption into a specific state. As time t→∞ the spin model will, with probability equal to 1, have all spins up, all spins down, or will oscillate between two antiferromagnetic (Néel) states. The time‐dependent correlation functions 〈s _{ i }(t)s _{ j }(t)〉 are also obtained.

Minimization of the energy functional of a one‐dimensional fermionic system in the large‐N limit
View Description Hide DescriptionA one‐dimensional system of N nonrelativistic fermions in the confining potential is studied in the large‐N limit where a classical limit appears.

Fractional approximations to the Bessel function J _{0}(x)
View Description Hide DescriptionA method to obtain fractional approximations to the Bessel function J _{0}(x) is reported here. This method improves a recently published one principally in that all the parameters are uniquely determined by linear equations. Our approximations give fairly good accuracy for all real, positive values. The maximum absolute error for the first‐degree approximation is about 0.0035, and for the fourth‐degree one, about 2.8×10^{−} ^{6}.