Volume 24, Issue 9, September 1983
Index of content:

Kac–Dynkin diagrams and supertableaux
View Description Hide DescriptionWe show the relation between Kac–Dynkin diagrams and supertableaux.

The SO(7) polynomial basis for symmetric representations
View Description Hide DescriptionA polynomial basis is derived for the symmetric irreducible representations of the group SO (7). The reduction of SO(7) into [SU(2)]^{3} is considered. The SO(7) generators not belonging to [SU(2)]^{3} are grouped into a bispinor vector, of which matrix elements are calculated. An explicit expression for the state vector is given.

A new approach to permutation group representations. II
View Description Hide DescriptionThe formula for the Yamanouchi matrix elements is rederived by the eigenfunction method in a simple fashion.

The geometric theory of deformation and linearization of Pfaffian systems and its applications to system theory and mathematical physics. I
View Description Hide DescriptionMany physical and engineering problems lead to the study of the qualitative and geometric properties of differential equations and their solutions as a function of exogeneous parameters. The theory of d e f o r m a t i o n o f g e o m e t r i c s t r u c t u r e s a n d p s e u d o g r o u p s initiated by K. Kodaira and D. C. Spencer deals with the common mathematical structure underlying these problems. This series of papers will adapt the work in the pure mathematics literature to the needs of the applications, with the emphasis on the theory of d e f o r m a t i o n s o f P f a f f i a n s y s t e m s. The applied area to be emphasized in this first part is the theory of nonlinear input–output systems. I will also present the abstract algebraic structure which seems to underlie the theory of Pfaffian systems, which I call C a r t a n–V e s s i o t f i l t r a t i o n s o f L i e a l g e b r a s.

The isospectral property for a family of non‐self‐adjoint operators
View Description Hide DescriptionWe prove the isospectral property for certain families of linear non‐self‐adjoint operators which play a role in inverse scatteringtheory for a class of nonlinear evolution equations of interest in physics. These include the sine‐Gordon and nonlinear Schrödinger equations.

A new class of integrable systems
View Description Hide DescriptionWe present a family of dynamical systems associated with the motion of a particle in two space dimensions. These systems possess a second integral of motion quadratic in velocities (apart from the Hamiltonian) and are thus completely integrable. They were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation(PDE) for the potential. A most important point is that the same PDE was derived through considerations on the analytic structure of the singularities of the solutions (‘‘weak‐Painlevé property’’).

Integrability of Hamiltonians with third‐ and fourth‐degree polynomial potentials
View Description Hide DescriptionThe weak‐Painlevé property, as a criterion of integrability, is applied to the case of simple Hamiltonians describing the motion of a particle in two‐dimensional polynomial potentials of degree three and four. This allows a complete identification of all the integrable cases of cubic potentials. In the case of quartic potentials, although our results are not exhaustive, some new integrable cases are discovered. In both cases the integrability is explicited by a direct calculation of the second integral of motion of the system.

A deformation of the general zero‐curvature equations associated to simple Lie algebras
View Description Hide DescriptionWe construct deformations and rational reductions for all the general zero‐curvature equations associated to simple complex Lie algebras [known as AKNS equations for sl(2, C)].

Higher‐order parabolic approximations to time‐independent wave equations
View Description Hide DescriptionA sequence of numerically tractable higher‐order parabolic approximations is derived for the reduced wave equation in an inhomogeneous medium. The derivation is motivated by a definition of waves propagating in a distinguished direction. For a homogeneous medium these definitions are exact and yield uncoupled, infinite‐order parabolic equations which are equivalent to the wave equation. The difficulty of obtaining higher‐order parabolic approximations for the elastic wave equation in an inhomogeneous medium is also discussed.

Simple waves in quasilinear hyperbolic systems. I. Theory of simple waves and simple states. Examples of applications
View Description Hide DescriptionThis paper presents a new method of construction of solutions to nonlinear, nonelliptic systems of partial differential equations and especially nonhomogeneous ones. These equations have been considered from the point of view of integral elements. In particular the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions have been studied. These classes consist of what is called simple waves and k waves (for homogeneous systems) and simple states (in the case of nonhomogeneous systems). They provide us with a possibility for a selection of simple integral elements from the set of all integral elements. Analyses have been performed using differential forms and Cartan theory of system in involution. The problem has been reduced to examining Pfaff forms. The Cauchy problem for Pfaff systems has been formulated and solved using the Riemann function. Some remarks concerning the notion of Bäcklund transformations for the case of k waves have been formulated. It is shown that, in contrast to simple wave, the simple state has no gradient catastrophy. The technique presented of constructing the solutions in form of simple states has been illustrated by the examples of Korteweg and de Vries and four‐dimensional Klein–Gordon, sine–Gordon, and Liouville equations. It has been shown that the known soliton equations are closely connected with the elliptical functions and especially with the P–Weierstrass functions.

Simple waves in quasilinear hyperbolic systems. II. Riemann invariants for the problem of simple wave interactions
View Description Hide DescriptionIn this paper a generalization of the Riemann invariant method to the case of a nonhomogeneous system of equations has been formulated. We have discussed in detail the necessary and sufficient conditions for the existence of Riemann invariants. We perform the analysis using the apparatus of differential forms and Cartan theory of systems in involution. The problem has been reduced to examining Pfaff forms. We have considered the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions depending on k arbitrary functions of one variable. These solutions can be interpreted physically as the interactions between k simple waves on a simple state. We have proven that, in the case of interaction of many simple waves described by Riemann invariants, a conservation law for the type and quantity of waves holds. It has been also shown that such a solution, resulting from the interaction of many simple waves propagating on the simple state, decay for a large time in an exact way into simple waves (of the same kind as those entering the interaction) on the state. The Cauchy problem for the nonlinear superposition of k‐sample waves has been formulated. A couple of theorems useful for this problem have been given in the Sec. III. The functorial properties of the system of equations determining Riemann invariants have been described. The last part of the work contains an analysis of some examples of the solutions of nonhomogeneous magnetohydrodynamic equations from the point of view of the method described above.

Generalized stochastic processes and continual observations in quantum mechanics
View Description Hide DescriptionWe give here a mathematically rigorous form to an earlier work by Barchielli, Lanz, and Prosperi, in which it was found that a generalized stochastic process describes the results of continual observations of the position of a quantum particle. With the help of Albeverio and Ho/egh‐Krohn’s theory of Feynman path integrals, we define the characteristic functional of this process and demonstrate that it possesses the necessary properties of normalization, continuity, and positive definiteness. An explicit calculation of the Feynman path integral which defines the functional allows an analysis of the process to be made.

Operators for the two‐dimensional harmonic oscillator in an angular momentum basis
View Description Hide DescriptionThe forms of the operators ν^{°}, ν, λ^{°}, λ, which enable one to write the Hamiltonian of the two‐dimensional isotropic harmonic oscillator in the form H=ℏω(2ν^{°}ν+λ^{°}⋅λ+1), are presented. Here ν^{°} and ν are, respectively, the raising and lowering operators for ν^{°}ν, the ‘‘radial’’ quantum number operator, while λ^{°} and λ are, respectively, the raising and lowering operators for M, the magnitude of the angular momentum operator. Corresponding to this decomposition of H in the angular momentum basis are the energy eigenvaluesE _{ k m }=ℏω(2k+‖m‖+1) with k=0, 1, 2, ⋅⋅⋅ and m=0, ±1, ±2, ⋅⋅⋅. Here k is a ‘‘radial’’ quantum number, and m is a ‘‘magnetic’’ quantum number. The commutation relations satisfied by the operators ν^{°}, ν, λ^{°}, and λ are also presented.

A time‐dependent Schrödinger equation
View Description Hide DescriptionWe reduce the solution of the Schrödinger equation with the potential U(r, t)=α_{1}(t) r ^{2}+α_{2}(t) x+α_{3}(t) y +α_{4}(t) z+α_{5}(t) to the solution of the Schrödinger equation for a free particle. The α_{ i }(t) are arbitrary functions of time. A generalization of this is also considered.

Lorentz covariance of an extended object in the tree approximation. II. Nonspherical object in 3+1 dimensions
View Description Hide DescriptionThis is the second in the series of the papers in which we investigate the Lorentz covariance of the extended object. In this paper we examine the covariance of the deformed object in 3+1 dimensions in the tree approximation. We construct the solution of the Euler equation, which is Lorentz covariant. In such a covariant solution, the variables associated with the rotational and the translational zero modes appear as classical quantum mechanical operators. Consequently the covariant solution has an intrinsic spin, in addition to the intrinsic quantum mechanical momenta. Then, at the end of this work we will show that such a covariant solution can be obtained also by quantizing a classical solution of the Euler equation, having extra variables signifying the center and the orientation of the deformed object. In the tree approximation, the energy–momentum and the relativistic angular momentum of the extended object ψ become pure classical quantum mechanical operators, having been integrated over the space. Then it is proven that such four‐momenta and angular momentum operators form a classical quantum mechanics presented in a relativistic manner. The center of mass of the extended object, often called collective coordinate, is shown to be made of these four‐momentum and angular momentum. This center of mass and the four‐momentum operators form a quantum mechanics presented in the conventional form.

An SL(2,C)‐invariant representation of the Dirac equation. II. Coulomb Green’s function
View Description Hide DescriptionThe Kepler problem for the Klein–Gordon type wave equation {Π_{μ}Π_{μ}+m ^{2}+i e σ⋅(E+i B)}φ=0, investigated earlier [J. Math. Phys. 2 3, 1179 (1982)] and proven to be equivalent to the conventional Dirac equation, is discussed. In this equation φ is a 2×1 Pauli spinor and σ_{ a }, a=1, 2, 3, are the usual 2×2 Pauli spin matrices. Quite simple expressions for the bound state Coulomb wavefunctions and for the Coulomb Green’s function are obtained by exploiting the concept of ‘‘coupling constant eigenfunction.’’ To facilitate the direct use of these simple expressions in Coulomb calculations, a stationary state perturbation theory appropriate for the Klein–Gordon type wave equation itself is described.

Infinitesimal symmetries and conserved currents for nonlinear Dirac equations
View Description Hide DescriptionThe complete algebras of infinitesimal symmetries for Dirac equations,Dirac equations with nonvanishing rest mass, and Dirac equations with a nonlinear term have been established. In addition, conserved currents associated to new symmetries have been constructed.