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Volume 35, Issue 11, November 1994

Long‐range scattering for three‐body Stark Hamiltonians
View Description Hide DescriptionThe existence and the asymptotic completeness of the Graf‐type modified wave operators for three‐body Stark Hamiltonians with long‐range potentials are proven under the condition that the electric field is sufficiently strong and the particles are accelerated with different acceleration.

A Lie algebraic study of some Schrödinger equations
View Description Hide DescriptionBound‐state solutions of several different Schrödinger equations are calculated rather efficiently using the methods of Lie algebra. In all cases considered, either of the algebras SO(3) or SO(2,1) provides a suitable framework, and there is no advantage to either choice. However, the choice of an appropriate realization of the generators is of greater significance, leading to a particularly simple solution whenever the Hamiltonian can be expressed as a linear function of the generators. In certain cases, the Hamiltonian can be expressed only as a bilinear function of the generators and only part of the bound‐state spectrum can be calculated analytically to yield a finite set of so‐called quasiexactsolutions. These may be interrelated by means of suitable ladder operators and it is possible, though by no means necessary, to adopt a particular finite‐dimensional representation of the underlying algebra. The present work emphasizes the role of similarity transformations, and the connection between the Lie method and other (generally variational) procedures which lead to the diagonalization of very large (in principle infinite‐dimensional) matrices.

Anosov actions on noncommutative algebras
View Description Hide DescriptionAn axiomatic framework for a quantum mechanical extension to the theory of Anosov systems is constructed, and it is shown that this retains some of the characteristic features of its classical counterpart, e.g., nonvanishing Lyapunov exponents, a vectorial K‐property, and exponential clustering. The effects of quantization are investigated on two prototype examples of Anosov systems, namely, the iterations of an automorphism of the torus (the ‘‘Arnold Cat’’ model) and the free dynamics of a particle on a surface of negative curvature. It emerges that the Anosov property survives quantization in the case of the former model, but not of the latter one. Finally, we show that the modular dynamics of a relativistic quantum field on the Rindler wedge of Minkowski space is that of an Anosov system.

A new lattice formulation of the continuum
View Description Hide DescriptionA new lattice formulation of the continuum is presented, which can be applied to quantum or classical field theories in any space–time dimension. The method is illustrated with an application to relativistic nontopological solitons. This approach also provides an effective means of removing the spurious lattice fermion degenerate solutions for the Dirac equation.

On transformations preserving the basis conditions of a spin structure group in four‐dimensional superstring theory in free fermionic formulation
View Description Hide DescriptionLet Ξ stand for a finite Abelian spin structure group of 4‐dimensional superstring theory in free fermionic formulation whose elements are 64‐dimensional vectors (spin structure vectors) with rational entries belonging to ]−1,1] and the group operation is the mod 2 entry by entry summation ⊕ of these vectors. Let B={b _{ i }, i=1,...,k+1} be a set of spin structure vectors such that b _{ i } have only entries 0 and 1 for any i=1,...,k, while b _{ k+1} is allowed to have any rational entries belonging to ]−1,1] with even N _{ k+1}, where N _{ k+1} stands for the least positive integer such that N _{ k+1} b _{ k+1}=0 mod 2. Let B be a basis of Ξ, i.e., let B generate Ξ, and let Λ_{ m,n } stand for the transformation of B which replaces b _{ n } by b _{ m }⊕b _{ n } for any m ≠ k+1, n ≠ 1, m ≠ n. It is proven that if B satisfies the axioms for a basis of spin structure group Ξ, then B’=Λ_{ m,n } B also satisfies the axioms. Since the transformations Λ_{ m,n } for different m and n generate all nondegenerate transformations of the basis B that preserve the vector b _{1} and a single vector b _{ k+1} with general rational entries, it can be concluded that the axioms are conditions for the whole group Ξ and not just conditions for a particular choice of its basis. Hence, these transformations generate the discrete symmetry group of four‐dimensional superstring models in free fermionic formulation. This is the physical meaning of the obtained result. The practical impact is that in searching for a realistic model in four‐dimensional superstring theory in free fermionic formulation it is enough to search only through different groups Ξ, which number dramatically less than the number of their different bases.

On the structure of Yang–Mills fields in compactified Minkowski space
View Description Hide DescriptionIn this work the Yang–Mills functional with gauge group SU(2) in compactified Minkowski space is reduced to a mechanical Lagrangian which enables one to find new classes of exact dual and nonself‐dual solutions. The technique used in the reduction allows a simple interpretation of meron‐type solutions. The formalism is then used to produce a good simple analytic approximation to static solutions of Skyrme’s field equation, relevant to low energy hadronic interactions.

Bounds for the exact matrix elements for radiation processes in hydrogenic atoms. II
View Description Hide DescriptionIn recent articles [H. E. Moses and R. T. Prosser, ‘‘Exact matrix elements for radiation processes in hydrogenic atoms,’’ J. Math. Phys. 33, 1878–1886 (1992); ‘‘Bounds for the Appell Function F _{2}’’] the authors have obtained uniform bounds for the exact matrix elements for elementary radiation processes involving the ground state in both nonrelativistic and relativistic hydrogenic atoms. Here these results are extended to obtain uniform bounds for the exact matrix elements involving other stationary states of these atoms. These results will be used in a sequel to bound the exact interaction operator in a study of radiation processes involving the hydrogen atom.

Vertex operators for physical states of bosonic string
View Description Hide DescriptionThe commutator of the Virasoro operator L _{ m } and general vertex operator V for the mass level L is explicitly calculated. By demanding that a physical vertex operator must be of conformal dimension J=1, a set of algebraic equations for determining the vertex operators corresponding to all physical states of bosonic strings is obtained. Explicit expressions for the first three mass levels are given.

Quantum chaos: An entropy approach
View Description Hide DescriptionA new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov–Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent statesentropy, is defined. It may be used to measurechaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent statesentropy of a quantum map as ℏ→0 is less than or equal to the KS‐entropy of the corresponding classical map. ‘‘Chaos umpire sits, And by decision more imbroils the fray By which he reigns: next him high arbiter Chance governs all.’’ John Milton, Paradise Lost, Book II

Potentials derived geometrically for symmetry scattering
View Description Hide DescriptionThe scattering interaction in symmetry scattering is related to certain local potentials. The resulting potentials are calculated from the density function. This function is connected with the geometry of the space where scattering takes place. The potentials admit a quantum mechanical supersymmetric treatment. The corresponding superpotential is proportional to the logarithmic derivative of the density function.

Global potentials for general relativistic flows of ideal charged fluids
View Description Hide DescriptionIt is shown that general relativistic nondissipative flows of a possibly charged one component fluid, submitted to the action of its own electromagnetic field, admit a representation in terms of global potentials. A corresponding variational principle is presented, from which the equations of motion for the fluid and the fields can be obtained.

Note on Maxwell’s equations in relativistically rotating frames
View Description Hide DescriptionIt is shown that the relativistic rotation transformation of Trocheris and Takeno restores the full Lorentz covariance of electrodynamics, lost in previous investigations using the Galilean rotation transformation along with the Frenet–Serret tetrad associated to the world line of a rotating observer of velocity v=ωΛr; it also recovers, at the approximation of small velocities, the results of the anholonomic approach corresponding to the use of the ‘‘instantaneous’’ Lorentz transformation of Galilean velocity v=ωr (in domains restricted by r<c/ω).

On the complete symmetry group of the classical Kepler system
View Description Hide DescriptionA rather strong concept of symmetry is introduced in classical mechanics, in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey. Accordingly, a ‘‘complete symmetry group’’ realization in mechanics must be endowed with the following two features: (1) the group acts freely and transitively on the manifold of all allowed motions of the system; (2) the given equations of motion are the onlyordinary differential equations that remain invariant under the specified action of the group. This program is applied successfully to the classical Kepler problem, since the complete symmetry group for this particular system is here obtained. The importance of this result for the quantum kinematic theory of the Kepler system is emphasized.

Infinite products, partition functions, and the Meinardus theorem
View Description Hide DescriptionThe Meinardus theorem provides an asymptotic value for the infinite‐product generating functions of many different partition functions, and thereby asymptotic values for these partition functions themselves. Under assumptions somewhat stronger than those of Meinardus a more specific version of the theorem is derived, then this is generalized from the spectrum of integers considered by Meinardus to infinite products constructed from an arbitrary spectrum. Included are infinite products that generate partition functions able to count states in the quantum theory of extended objects. There emerges a quite general and explicit method (of which only limited details are given here) for deriving the asymptotic growth of the state density function of multidimensional quantum objects. One finds a universal geometry‐independent leading behavior, with nonleading geometry‐dependent corrections that the method is capable of providing explicitly.

Dynamical stability analysis of strong/weak wave collapses
View Description Hide DescriptionThe dynamical stability of self‐similar wave collapses is investigated in the framework of the radially symmetric nonlinear Schrödinger equation defined at space dimensions exceeding a critical value. The so‐called ‘‘strong’’ collapse, for which the mass of a collapsing solution remains concentrated near its central self‐similar core, is shown to be characterized by an unstable contraction rate as time reaches the collapse singularity. By contrast with this latter case, a so‐called ‘‘weak’’ collapse, whose mass dissipates into an asymptotic tail, is proven to contain a stable attractor from which a physical self‐similar collapse may be realized.

The (N,M)th Korteweg–de Vries hierarchy and the associated W‐algebra
View Description Hide DescriptionA differential integrable hierarchy, which is called the (N,M)th Korteweg–de Vries (KdV) hierarchy, whose Lax operator is obtained by properly adding M pseudo‐ differential terms to the Lax operator of the Nth KdV hierarchy is discussed herein. This new hierarchy contains both the higher KdV hierarchy and multifield representation of the Kadomtsev–Petviashvili (KP) hierarchy as subsystems and naturally appears in multimatrix models. The N+2M−1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are local and polynomial . Each Poisson structure generates an extended W _{1+∞}‐ and W _{∞}‐algebras, respectively. W(N,M) is called the generating algebra of the extended W _{∞}‐algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual W _{ N }‐algebra. It is shown that there exist M distinct reductions of the (N,M)th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N+M)th KdV hierarchy. Correspondingly the W(N,M)‐algebra is reduced to the W _{ N+M }‐algebra. The dispersionless limit of this hierarchy and the relevant reductions are studied in detail.

Rational solutions of the Kadomtsev–Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rational solution
View Description Hide DescriptionA new approach to the construction of the rational solutions to the hierarchy of the Kadomtsev–Petviashvili equation is presented. The generalization of the ‘‘superposition formula’’ is found to permit the construction of a new general solution from some partial ones. The features of the general polynomial factorization and the types of scattering in the many‐body Calogero–Moser problem are investigated.

Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain
View Description Hide DescriptionIt is desirable for one to be able to construct separation variables for systems with SL(N) R matrices or Lax pairs. This article relates and extends the two known methods of construction. The Sklyanin’s functional Bethe ansatz (FBA) [Commun. Math. Phys. 150, 181–191 (1992)] was previously only known for SL(2) and SL(3) (and associated) R matrices. The algebraic geometric method of Adams, Harnad, and Hurtubise [Commun. Math. Phys. 155, 385–413 (1993)] has been shown to work for SL(N) but up until now only for linear Poisson brackets. In this article Sklyanin’s program is advanced by giving the FBA for certain systems with SL(N) R matrices. This is achieved by constructing rational functionsA(u) and B(u) of the matrix elements of T(u), so that, in the generic case, the zeros x _{ i } of B(u) are the separation coordinates and the P _{ i }=A(x _{ i }) provide their conjugate momenta. It is shown that the separation variables thus defined are the same as those given (as the simultaneous solutions of several equations in two variables) by the other method. The crucial calculation of Adams et al. of the commutation relations is also adapted to the case of quadratic Poisson brackets. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed. No knowledge of algebraic geometry is required.

Calogero–Moser hierarchy and KP hierarchy
View Description Hide DescriptionIdentification of the hierarchy of Calogero–Moser dynamical systems with the dynamical systems of poles of certain solutions to the KP hierarchy is achieved.

The Kaup–Broer system and trilinear odes of Painlevé type
View Description Hide DescriptionThe Kaup–Broer hierarchy can be considered as a certain reduction of the KP hierarchy. Upon introducing the τ‐function the Kaup–Broer system can be formulated in trilinear form giving rise to Wronskian solutions. In this paper several odes of the Painlevé type obtained through reductions of the Kaup–Broer system are considered. Those odes take a trilinear form which allows a linearization.