Robert G. Littlejohn

Hamiltonian formulation of guiding center motion

A Hamiltonian theory of guiding center motion which uses rectangular coordinates in physical space and noncanonical coordinates in phase space is presented. The averaging methods preserve two important features of Hamiltonian systems, viz., conservation of energy (for time‐independent fields) and Liouville’s theorem. These features are sacrificed by the traditional averaging methods. The methods also relieve much of the burden of higher order perturbation calculations, and the drift equations for fully electromagnetic fields are extended to one higher order than they have been known in the past. The first correction to the relativistic magnetic moment is also calculated. Many applications are anticipated, both to single particle motion and to kinetic theory.

A guiding center Hamiltonian: A new approach

A Hamiltonian treatment of the guiding center problem is given which employs noncanonical coordinates in phase space. Separation of the unperturbed system from the perturbation is achieved by using a coordinate transformation suggested by a theorem of Darboux. As a model to illustrate the method, motion in the magnetic field B=B (x,y) ? is studied. Lie transforms are used to carry out the perturbation expansion.

Hamiltonian perturbation theory in noncanonical coordinates

The traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented in a way which clearly displays their differential‐geometric foundations. These are then generalized to the case of noncanonical in phase space. In the new method the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p dq−Hdt. The perturbation analysis is applied to this entire form, in all of its components.

A general framework for discrete variable representation basis sets

A framework for discrete variable representation (DVR) basis sets is developed that is suitable for multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators in a central role and integrates semiclassical and phase space concepts into the basic framework. Rates of convergence of basis set expansions are emphasized, and it is shown that the DVR method gives exponential convergence, assuming conditions of analyticity and boundary conditions are met. A discussion of nonorthogonal generalizations of DVR functions is presented, in which it is shown that projected δ-functions and interpolating functions form a biorthogonal basis. It is also shown that one of the generalized DVR proposals due to Szalay [J. Chem. Phys. 105, 6940 (1996)] gives exponential convergence.

The Van Vleck formula, Maslov theory, and phase space geometry

The Van Vleck formula is an approximate, semiclassical expression for the quantum propagator. It is the starting point for the derivation of the Gutzwiller trace formula, and through this, a variety of other expansions representing eigenvalues, wave functions, and matrix elements in terms of classical periodic orbits. These are currently among the best and most promising theoretical tools for understanding the asymptotic behavior of quantum systems whose classical analogs are chaotic. Nevertheless, there are currently several questions remaining about the meaning and validity of the Van Vleck formula, such as those involving its behavior for long times. This article surveys an important aspect of the Van Vleck formula, namely, the relationship between it and phase space geometry, as revealed by Maslovs theory of wave asymptotics. The geometrical constructions involved are developed with a minimum of mathematical formalism.

Semiclassical mechanics of the Wigner 6j-symbol

The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems to explore the geometrical issues surrounding the Ponzano?Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano?Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson) and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol.

Semiclassical Structure of Trace Formulas

Trace formulas provide the only general relations known connecting quantum mechanics with classical mechanics in the case that the classical motion is chaotic. In particular, they connect quantal objects such as the density of states with classical periodic orbits. In this paper, several trace formulas, including those of Gutzwiller, Balian and Bloch, Tabor, and Berry, are examined from a geometrical standpoint. New forms of the amplitude determinant in asymptotic theory are developed as tools for this examination. The meaning of caustics in these formulas is revealed in terms of intersections of Lagrangian manifolds in phase space. The periodic orbits themselves appear as caustics of an unstable kind, lying on the intersection of two Lagrangian manifolds in the appropriate phase space. New insight is obtained into the Weyl correspondence and the Wigner function, especially their caustic structures.

Differential forms and canonical variables for drift motion in toroidal geometry

Differential forms provide a simple and powerful computational tool for finding canonical variables for drift motion in toroidal geometry. The method is easily applied to relativistic drift motion.

Semiclassical analysis of Wigner 3j-symbol

We analyse the asymptotics of the Wigner 3j -symbol as a matrix element connecting eigenfunctions of a pair of integrable systems, obtained by lifting the problem of the addition of angular momenta into the space of Schwinger’s oscillators. A novel element is the appearance of compact Lagrangian manifolds that are not tori, due to the fact that the observables defining the quantum states are noncommuting. These manifolds can be quantized by generalized Bohr–Sommerfeld rules and yield all the correct quantum numbers. The geometry of the classical angular momentum vectors emerges in a clear manner. Efficient methods for computing amplitude determinants in terms of Poisson brackets are developed and illustrated.

Bessel discrete variable representation bases

Discrete variable representation (DVR) basis sets on the radial half-line, based on Bessel functions, are presented. These are Hankel transforms of the eigenfunctions of the particle in a spherical box in k space, but there is no box or bound on the radial variable r. The grid points extend to infinity on the r axis. The DVR functions are exactly orthonormal and exactly satisfy the interpolation properties usually associated with DVR functions. The exact matrix elements of the kinetic energy are computed, and the use of the Bessel DVR functions in radial eigenvalue problems is illustrated. The phase space or semiclassical interpretation of the Bessel DVR functions is presented, and variations on these functions, corresponding to alternative boundary conditions in k space, are discussed. An interesting feature of Bessel DVR functions is that they are based on a finite basis representation that is continuously infinite.