Volume 25, Issue 11, November 1984
Index of content:

Reduction formulas for higher‐order indices and anomaly
View Description Hide DescriptionReduction formulas for {1^{ k }} and {k} of SU(N) are derived which are useful for obtaining higher‐order indices. We also touch on the anomaly in higher‐dimensional theories.

Space groups and their isotropy subgroups
View Description Hide DescriptionThe theory of space groups and their subgroups is related to that of their irreducible representations. Several important theorems concerning the determination of isotropy subgroups are proved. In particular, we show that the translation subgroups of isotropy subgroups are in one‐to‐one correspondence with certain subsets, called ‘‘substars,’’ of the ‘‘star’’ characterizing the irreducible representation of the space group.

Classification of systems of nonlinear ordinary differential equations with superposition principles
View Description Hide DescriptionA system of n first‐order nonlinear ordinary differential equationsẋ(t)=f(x,t) is said to admit a superposition principle if its general solution can be written as a function of a finite number m of particular solutions and n constants. Such a system can be associated with the nonlinear action of a Lie groupG on a space M. We show that ‘‘indecomposable’’ systems of ODE’s with supersposition principles are obtained if and only if the Lie algebrasL _{0}⊆L, corresponding to the isotropy group H of a point and the group G, respectively, define a transitive primitive filtered Lie algebra (L,L _{0}). Using known results from the theory of transitive primitive Lie algebras we deduce that L _{0} must be a maximal subalgebra of L and that G must be an affine group, a simple Lie group, or the direct product of two identical simple Lie groups. Affine groups lead to linear equations, the other types to nonlinear equations with polynomial or rational nonlinearities. Equations corresponding to the classical complex Lie algebras are worked out in detail.

A new set of Euler angles for the generalized Lorentz group
View Description Hide DescriptionA new set of Euler angles for the generalized Lorentz group O^{+}( p,n−p) are defined, which turn out to be much simpler than the ones defined in a couple of earlier papers, and have the useful property that each factor in the factorization of a general element itself belongs to the same group.

Structure and representations of the hyperoctahedral group
View Description Hide DescriptionIn this paper, the main properties of the symmetry group of the n‐dimensional cube are reviewed and formulated with respect to possible applications in lattice theories. The connection between the hyperoctahedral group W _{ n } and the orthogonal group O(n) is investigated by means of the canonical representation.

Color analysis, variational self‐adjointness, and color Poisson (super)algebras
View Description Hide DescriptionAfter stating some facts concerning the calculation with ‘‘color variables’’ we cite some recent results of the author with respect to the color analytic extension of variational principles, self‐adjointness, and Heisenberg commutation relations. As an apparent novelty, we then present the color analytic version of the Hamiltonian formalism including the construction of color Poisson brackets leading to a color (super)algebra with color derivation property.

Necessary conditions for a unique solution to two‐dimensional phase recovery
View Description Hide DescriptionIn this paper we show that although in one dimension multiplicity of solutions to the phase reconstruction problem presents a serious problem, in two or more dimensions multiplicity is pathologically rare. We derive from a given solution pair (g,G) necessary conditions for the existence of alternative solution pairs (h,H), and a characterization of their form. The mathematical tools employed are from the theory of functions of two complex variables.

Bounds for the continuation of perturbative results to the spectral region
View Description Hide DescriptionThe problem of analytic continuation to the boundary of the holomorphy domain from both continuous and discrete interior sets has recently been the subject of detailed analyses. This problem is important in phenomenological applications but is also of interest in theoretical calculations, e.g., in attempting to evaluate the parameters of resonances or other nonperturbative effects in QCD. Because of the inherent instability of the continuation problem it is necessary to introduce additional criteria—which should be physically based—to select the right continuation function. In this paper, the results thus obtained for continuation from a continuum are examined for stability, and bounds are derived for the errors on the boundary in terms of the uncertainty of the input data. The procedure is shown to be stable in the sense that these bounds tend to zero as the data errors go to zero.

Commutants of a family of operators on a partial inner product space
View Description Hide DescriptionWe consider all different possible definitions of commutants and bicommutants for an x‐invariant family of operators on a partial inner product space. We investigate their behavior with respect to the weak topology and we describe the situation when all commutants (resp. all bicommutants) coincide.

Holonomy groups, sesquidual torsion fields, and SU(8) in d=11 supergravity
View Description Hide DescriptionThe torsion and its curl form an anti‐self‐dual SO(8) tensor field F _{ A B C D }( y). The Maxwell equations are solved if this tensor is covariantly constant, while the Einstein equations are solved if it satisfies an algebraic relation at the origin. Such F _{ A B C D } are found as the kernel of the holonomy group in the 35 representation of SO(8) and they extend the SO(8) of the round S _{7} to SU(8).

Liouville and Painlevé equations and Yang–Mills strings
View Description Hide DescriptionStringlike solutions of the self‐dual Yang–Mills equations (dimensionally reduced to R ^{2}) are sought. A multistring A n s a t z results in the sinh–Gordon and Liouville equations. According to a general theorem, the solutions must be either real and singular and have infinite action, or complex and nonsingular, with zero action. In the Liouville case, explicit arbitrarily separated n‐string solutions of both classes are given. The magnetic flux for these solutions is found to be the Chern class of a Kaehler manifold, and it consequently assumes quantized values 4πn/e. The axisymmetric version of the sinh–Gordon is solved by the third Painlevé transcendent P _{3}, using the results on P _{3} by Wu e t a l. [Phys. Rev. B 1 3, 316 (1976)] and McCoy e t a l. [J. Math. Phys. 1 8, 10 (1977)]. The axisymmetric case can be cast into the Ernst equation framework for the generation of further solutions. In the Appendix, the Euclideanized Ernst equation is shown to give self‐dual Gibbons–Hawking gravitational instantons.

First integrals via polynomial canonical transformations
View Description Hide DescriptionMaharatna, Dutt, and Chattarji [J. Math. Phys. 2 0, 2221 (1979)] discussed the use of time‐dependent canonical transformations for the determination of first integrals for time‐dependent Hamiltonian systems. One particular proposal that successive time‐dependent polynomial canonical transformations will enable first integrals to be found for a wider variety of time‐dependent polynomial Hamiltonians than can be obtained using time‐dependent linear canonical transformations is shown to be not true for the paradigm which they selected. It is suggested that their ansatz is ill‐founded in general.

Factorization of the 2×2 matrix recursion operator of the coupled KdV equation
View Description Hide DescriptionThe recursion operator for the infinitesimal transformations about solutions of the coupled KdV equation is a 2×2 matrix whose elements are of the fourth order. This formidable looking operator is written as the product of four 2×2 matrix operators whose elements are of the first order. Auxiliary functions introduced to factorize the recursion operator lead to the scattering equation for the equation. The factorization of the recursion operator for the sine–Gordon equation is also presented.

Inverse scattering for geophysical problems. II. Inversion of acoustical data
View Description Hide DescriptionAn algorithm is given for finding the density and bulk modulus (refraction coefficient) of an inhomogeneity from the knowledge of the scattered field on the surface of the earth for all positions of the source and receiver on this surface and for two arbitrary fixed frequencies in the Born approximation. An alternative inversion method using the low‐frequency data is also given.

Electrodynamics of memory‐dependent nonlocal elastic continua
View Description Hide DescriptionBalance laws and constitutive equations are given for elastic continua with memory of past motions and electromagnetic fields. Nonlinear, finite‐linear, and linear constitutive equations are obtained and restricted by the second law of thermodynamics. Memory‐dependent nonlocal piezoelectricity, piezomagnetism, heat and electric conduction,viscoelasticity, and other allied physical phenomena are in the domain of the general theory. The theory is applied to discuss infrared dispersion and lattice vibrations, natural optical activity, anomalous skin effect, and superconductivity, indicating the power and the potential of the nonlocal theory.

Proof of Regge analyticity for power law potentials
View Description Hide DescriptionThe l‐plane analyticity of Schrödinger energy levels E _{ n }(l) for power law potentials V(r)=r ^{α}, for α>2 has been proved by using the Kato–Rellich perturbation theory for linear operators.

Functions analytic on the half‐plane as quantum mechanical states
View Description Hide DescriptionA transform between the state space of one‐dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half‐plane is constructed. It gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple. Generalizations to higher dimensions are also discussed.

Quantum systems with time‐dependent harmonic part and the Morse index
View Description Hide DescriptionA simple relation between two quantum systems with a time‐dependent, respectively, time‐independent, harmonic part is established. Using this we give a computation, valid for all times, of the Green’s functions of the time‐dependent harmonic oscillator with and without a perturbation of the type g/x ^{2}. The asymptotic expansion of the wave function in powers of Planck’s constant is discussed using a new representation of the Morse index.

Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation
View Description Hide DescriptionLet u be a nontrivial, smooth solution to i u _{ t }=Δu−‖u‖^{ p−1} u. If n=1 and 2<p≤3, then there does not exist any finite energy free solution v such that ∥u(t)−v(t)∥_{2}→0 as t→+∞. This extends a theorem of Strauss in which the same result was proved for 1<p≤2.

Anisotropic fluids and conformal motions in general relativity
View Description Hide DescriptionWe study the consequences of the existence of a one‐parameter group of conformal motions for anisotropic matter, in the context of general relativity. It is shown that for a class of conformal motions (special conformal motions), the equation of state is uniquely determined by the Einstein equations. For spherically symmetric and static distributions of matter we found two analytical solutions of the Einstein equations which correspond to isotropic and anisotropic matter, respectively. Both solutions can be matched to the Schwarzschild exterior metric and possesses positive energy density larger than the stresses, everywhere within the sphere.