Showing papers on "Lie group published in 1983"

On the controllability of quantum‐mechanical systems

Abstract: The systems‐theoretic concept of controllability is elaborated for quantum‐mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space H of the system. The Schrodinger equation for a quantum object influenced by adjustable external fields provides a state‐evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain Dω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite‐dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of Dω with a suitably defined finite‐dimensional submanifold of the unit sphere SH in H. Several soluble examples are presented to illuminate th...

Nonlinear gyrokinetic equations

Abstract: Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, only electrostatic fluctuations in slab geometry are considered; however, there is a straightforward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and several limiting forms are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev can ony be derived by an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry–Horton and Hasegawa–Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed. The resulting theory is very similar in content to the recent work of Lee. However, the systematic nature of our derivation provides considerable insight into the structure and interpretation of the equations.

Potential Scattering, Transfer Matrix, and Group Theory

Abstract: Both bound and scattering states of the P\"oschl-Teller potential are shown to be connected with unitary representations of certain groups. A family of periodic potentials, and their associated transfer matrices and band structure, can also be obtained from group theory and reduce to the above potential when the real period approaches infinity. These results suggest that the algebraic approach used for treating bound-state problems can be extended to scattering and band-structure problems.

The Method of Differential Approximation

Abstract: I am very glad that this book is now accessible to English-speaking scientists During the three years following the publication of the original Russian edition, the method of differential approximation has been rapidly expanded and unfortunately I was unable to incorporate into the English edition all of the material which whould have reflected its present state of development Nevertheless, a considerable amount of recently obtained results have been added and the bibliography has been enlarged accordingly, so that the English edition is one third longer than the Russian original Mathematical rigorousness is a basic feature of this monograph The reader should therefore be familiar with the theory of partial differential equations and difference equations Some knowledge of group theory as applied to problems in physics, especially the theory of Lie groups, would also be useful The treatment of the approximation of gas dynamic equations focuses on the question of how to characterize the typical features of difference equations on the basis of the related differential approximation, which can be discussed using the fully developed theory of partial differential equations

Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups

Abstract: Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groupstion commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods

Abstract: Lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general Hamiltonian systems. A procedure is presented for the computation of nonlinear effects of arbitrarily high degree, and explicit formulas are given through effects of degree 5. Expected applications include accelerator design, charged particle beam and light optics, other problems in the general area of nonlinear dynamics, and, perhaps, with suitable modification, the area of S‐matrix expansions in quantum field theory.

Kronecker products for compact semisimple Lie groups

Abstract: A review is given of the application of S-function techniques to the evaluation of Kronecker products of irreducible representations of compact semisimple Lie groups. Explicit formulae are derived for all irreducible representations of all such groups. Recent developments involving composite Young diagrams are brought to fruition and the vexed problem of SO2k is dealt with completely. New branching rules for the classical groups are given in an appendix. These are exploited in the evaluation of Kronecker products by means of a technique which is applied to both classical and exceptional groups. A discussion is made of various modification rules which are needed to express the final results in standard form.

Teleparallelism-A viable theory of gravity?

Abstract: The teleparallelism theory of gravity is presented as a constrained Poincar\'e gauge theory. Arguments are given in favor of a two-parameter family of field Lagrangians quadratic in torsion. The inclusion of a "parity violating" term in the field Lagrangian avoids difficulties with the initial-data problem, recently discussed in the literature. Several new exact solutions of the corresponding teleparallelism theory give considerable insight into its physical consequences. The resulting general field equations are analyzed in the weak-field approximation excluding ghosts and tachyons. The physical meaning of the six additional components of the tetrad field (as compared with the metric) appears naturally from our theory and is made clear.

On collective complete integrability according to the method of Thimm

Abstract: Let G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

Covariance and geometrical invariance in *quantization

Abstract: Several notions of invariance and covariance for * products with respect to Lie algebras and Lie groups are investigated. Some examples, including the Poincare group, are given. The passage from the Lie‐algebra invariance to the Lie‐group covariance is performed. The compact and nilpotent cases are treated.

A Generalization of Casselman’s Submodule Theorem

Abstract: Let Gℝ be a real reductive Lie group, g;ℝ its Lie algebra. Let M be an irreducible Harish-Chandra module. Using some fine analytic arguments, based on the study of asymptotic behavior of matrix coefficients, Casselman has proved that M can be imbedded into a principal series representation [2,3].

Group properties and new solutions of Navier-Stokes equations

Abstract: Using the machinery of Lie theory (groups and algebras) applied to the Navier-Stokes equations a number of exact solutions for the steady state are derived in (two) three dimensions. It is then shown how each of these generates an infinite number of time-dependent solutions via (three) four arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

On the Existence of Exotic Banach-Lie Groups

Abstract: Let X be a real or complex Banach space, and let G be a topological group. By a representation of G in X we mean a homomorphism of G into GL(X) the group of all automorphisms ofX. A representation T : G ~ G L ( X ) is called weakly (resp. strongly, uniformly) continuous, when it is continuous in the weak (resp. strong, uniform) topology on GL(X). A representation T : G ~ G L ( X ) is called unitary, when X is a Hilbert space, and all the operators T(g), g~ G, are unitary. For unitary representations the notions of weak and strong continuity coincide. In this paper we shall frequently treat vector spaces as additive groups, and topological vector spaces (especially normed spaces) as additive topological groups. Thus, for example, if E is a normed space, we shall speak of the topological group E, of a subgroup K C E, or of a topological quotient group E/K, without special explanations. An abelian topological group is called exotic, if it does not admit any nontrivial strongly continuous unitary representations, and strongly exotic, if it does not admit any non-trivial weakly continuous representations in Hilbert spaces. An example of an exotic group was given by Herer and Christensen in [4]. The group in the mentioned example was a Polish vector space. In the present paper we exhibit the existence of strongly exotic Banach-Lie groups. More precisely, our aim is to prove the following fact:

Convexity and Loop Groups

Abstract: The purpose of this paper is to extend certain convexity results associated with compact Lie groups to an infinite-dimensional setting, in which the Lie group is replaced by the corresponding loop group. To recall the finite-dimensional results which we shall generalize let G be a simply connected, compact Lie group, T a maximal torus of G and W its Weyl group. Consider the adjoint action of G on its Lie algebra L(G) and fix a G-invariant metric on L(C) so that we can define orthogonal projection. A result of Kostant [8] describes the images of the G-orbits in L(G) under the orthogonal projection onto L(T). To state it, recall that such G-orbits correspond to W-orbits in L(T). Then Kostant's result is:

The unitary spectrum for real rank one groups

Abstract: Let G be a connected real semi-simple Lie group with finite center. One of the main problems in harmonic analysis is to determine the unitary spectrum of G. In this paper we treat this question in the case when real rank of G is I. Although the answer was known for G of classical type previously ([2], [5], [7], [15]), we have redone this work sometimes giving simpler arguments. With the arbitrary rank case in mind we have tried to deal with the problem of classifying unitary representations in as systematic a way as possible pro- ceeding by induction on the dimension of G. We have concentrated on the case when G is linear and has a compact Cartan subgroup, the other cases being known already. As an application we also give a list of the unitary representations that contribute to the L2-index formula for the Dirac operator with coefficients. As is apparent from the calculations in [19], this is intimately connected with one of our main techniques for determining the unitary spectrum, the Dirac in- equality. We will deal with some related problems in a future paper.

Fourier Transforms of Orbits of the Coadjoint Representation

Abstract: Let G be a compact Lie group with Lie algebra g. Let 0 ⊂ g* be an orbit of G under the coadjoint representation of maximal dimension 2n. For f ∈ 0, we denote by G(f) the stabilizer of f and t = ℊ(f) the Lie algebra of G(f). Let W be the Weyl group of (g, t). Recall that 0 is a symplectic manifold with a canonical 2-form σ.

Hamiltonian structure of Sato's hierarchy of KP equations and a coadjoint orbit of a certain formal lie group

Abstract: A Hamiltonian operator related to Sato's hierarchy of KP equations is derived. The KP equation is regarded as the Hamiltonian equation of motion on a coadjoint orbit of a certain formal Lie group.