Volume 38, Issue 1, January 1997
Index of content:

Nonnegative Feynman–Kac kernels in Schrödinger’s interpolation problem
View Description Hide DescriptionThe local formulations of the Markovian interpolating dynamics, which is constrained by the prescribed inputoutput statistics data, usually utilize strictly positive Feynman–Kac kernels. This implies that the related Markovdiffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process. We discuss an extension of the framework to encompass singular potentials and associated nonnegative Feynman–Kactype kernels. It allows us to deal with a class of continuous interpolations admitted by general nonnegative solutions of the Schrödinger boundary data problem. The resulting nonstationary stochastic processes are capable of both developing and destroying nodes (zeros) of probability densities in the course of their evolution, also away from the spatial boundaries. This observation conforms with the general mathematical theory (due to M. Nagasawa and R. Aebi) that is based on the notion of multiplicative functionals, extending in turn the well known Doob’s htransformation technique. In view of emphasizing the role of the theory of nonnegative solutions of parabolic partial differential equations and the link with “Wiener exclusion” techniques used to evaluate certain Wiener functionals, we give an alternative insight into the issue, that opens a transparent route towards applications.

Scattering by singular potentials. B. Varying the linear parameters
View Description Hide DescriptionIn part A a convergent WKB (WentzelKramersBrillouin) expansion has been developed for the wave function of Schrödinger scattering by highly singular potentials. In part B we look for the termbyterm dependence of this series on energy, coupling constant and orbital angular momentum. While we vary these parameters each point of fixed value of the new dimensionless radial coordinate invariably belongs either to the exponential or the trigonometric WKB region. By this technique, the series was proven to become exact even after the 2nd term both in the short wavelength, the strong coupling or the high partial wave limits.

NonAbelian topological mass generation in four dimensions
View Description Hide DescriptionWe study the topological mass generation in the 4 dimensional nonAbelian gauge theory, which is the extension of Allen et al.’s work in the Abelian theory. It is crucial to introduce a one form auxiliary field in constructing the gauge invariant nonAbelian action which contains both the one form vector gauge field and the two form antisymmetric tensor field . As in the Abelian case, the topological coupling , where is the field strength of , makes the transmutation among and possible, and consequently we see that the gauge field becomes massive. We find the BRST/antiBRST transformation rule using the horizontality condition, and construct a BRST/antiBRST invariant quantum action.

Geometrical approach to inverse scattering for the Dirac equation
View Description Hide DescriptionThe highenergylimit of the scattering operator for multidimensional relativistic dynamics, including a Dirac particle in an electromagnetic field, is investigated by using timedependent, geometrical methods. This yields a reconstruction formula, by which the field can be obtained uniquely from scattering data.

Lattice topological field theory on nonorientable surfaces
View Description Hide DescriptionThe lattice definition of the twodimensional topological quantum field theory [Fukuma et al., Commun. Math. Phys. 161, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a onetoone correspondence between real associative *algebras and the topological state sum invariants defined on such surfaces. The partition and npoint functions on all twodimensional surfaces (connected sums of the Klein bottle or projective plane and gtori) are defined and computed for arbitrary *algebras in general, and for the group ringA=R[G] of discrete groups G, in particular.

Ergodic properties of quantized toral automorphisms
View Description Hide DescriptionWe study the ergodic properties for a class of quantized toral automorphisms, namely the cat and Kronecker maps. The present work uses and extends the results of Klimek and Leśniewski [Ann. Phys. 244, 173–198 (1996)]. We show that quantized cat maps are strongly mixing, while Kronecker maps are ergodic and nonmixing. We also study the structure of these quantum maps and show that they are effected by unitary endomorphisms of a suitable vector bundle over a torus. This allows us to exhibit explicit relations between our Toeplitz quantization and the semiclassical quantization of cat maps proposed by Hannay and Berry [Physica D 1, 267–290 (1980)].

Displacementoperator squeezed states. I. Timedependent systems having isomorphic symmetry algebras
View Description Hide DescriptionIn this paper we use the Lie algebra of spacetime symmetries to construct states which are solutions to the timedependent Schrödinger equation for systems with potentials We describe a set of numberoperator eigenstates states, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, We prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters. Specific examples, all corresponding to choices of and having isomorphic Lie algebras, will be dealt with in the following paper (II).

Displacementoperator squeezed states. II. Examples of timedependent systems having isomorphic symmetry algebras
View Description Hide DescriptionIn this paper, results from the previous paper (I) are applied to calculations of squeezed states for such wellknown systems as the harmonic oscillator, free particle, linear potential, oscillator with a uniform driving force, and repulsive oscillator. For each example, expressions for the expectation values of position and momentum are derived in terms of the initial position and momentum, as well as in the  and in the representations described in I. The dependence of the squeezedstate uncertainty products on the time and on the squeezing parameters is determined for each system.

Construction of a complete set of states in relativistic scattering theory
View Description Hide DescriptionThe space of physical states in relativistic scattering theory is constructed, using a rigorous version of the Dirac formalism, where the Hilbert space structure is extended to a Gel’fand triple. This extension enables the construction of “a complete set of states,” the basic concept of the original Dirac formalism, also in the cases of unbounded operators and continuous spectra. We construct explicitly the Gel’fand triple and a complete set of “plane waves”—momentum eigenstates—using the group of space–time symmetries. This construction is used (in a separate article) to prove a generalization of the Coleman–Mandula theorem to higher dimension.

Generalization of the Coleman–Mandula theorem to higher dimension
View Description Hide DescriptionThe Coleman–Mandula theorem, which states that space–time and internal symmetries cannot be combined in any but a trivial way, is generalized to an arbitrarily higher spacelike dimension. Prospects for further generalizations of the theorem (spacelike representations, larger timelike dimension, infinite number of particle types) are also discussed. The original proof relied heavily on the Dirac formalism, which was not well defined mathematically at that time. The proof given here is based on the rigorous version of the Dirac formalism, based on the theory of distributions. This work also serves to demonstrate the suitability of this formalism for practical applications.

Geometrical stochastic control and quantization
View Description Hide DescriptionA class of diffusion processes controlled by the geometry of the manifold on which they evolve is considered. The kinetic energy of such diffusions is shown to be a geometric object expressed in terms of the curvature and torsion tensors. This gives rise to an action functional leading to a variational principle, from which a nontrivial critical geometry with nonvanishing torsion emerges. The resulting criticality condition is related to the Schrödinger equation in a manner that reproduces the features of Nelson’s stochastic approach to quantum mechanics.

Threedimensional electromagnetic inverse scattering for biisotropic dispersive media
View Description Hide DescriptionThe timedomain inverse problem of determining the threedimensional parameter functions for the biisotropic dispersive medium is considered. Maxwell’sequations are rewritten in terms of the tangential fields. Time domain wavesplitting of Maxwell’sequations is applied to the total field that is generated by a dipole exterior to the scattering medium. The structure of the electromagnetic fields is analyzed, and the transport equations are given. The reconstruction condition is derived for a layerstripping approach.

The Kowalewski top: A new Lax representation
View Description Hide DescriptionThe 2×2 monodromy matrices for the Kowalewski top on the Lie algebrase(3), so(4), and so(3,1) are presented. The corresponding quadratic Rmatrix structure is the dynamical deformation of the standard Rmatrix algebras. Some tops and Toda lattices related to the Kowalewski top are discussed.

Hamiltonian structure for degenerate AKNS systems
View Description Hide DescriptionThere is a family of degenerate AKNS systems for which the full theory of generic AKNS systems does not directly extend. The linear space of potentials still has a natural Poisson structure, but the scattering method used by Beals and Sattinger to show complete integrability for the generic AKNS systems fails for the degenerate case. A Poisson structure is not induced on the scattering side as in the generic case. As a consequence, the problem of complete integrability for degenerate AKNS systems still is an open question. In addition, contrary to the generic case, the Lax pair gives flows for degenerate integrable systems that are nonlocal. In general, they do not exist, and they are no longer linear on the scattering side. Necessary conditions for their existence and for linear evolution in the scattering side are found.

An inverse scattering transform for the MKdV equation with nonvanishing boundary value
View Description Hide DescriptionThe MKdV equation of normal dispersion with nonvanishing boundary value is solved by the inverse scattering transform method. An affine parameter is introduced to avoid doublevalued functions of the usual spectral parameter. In terms of it the inverse scattering transform is performed and the inverse scattering equation of Zakharov–Shabat form as well as of Marchenko form is derived. Dark multisoliton solutions are found formally by means of the Binet–Cauchy formula. The asymptotic behaviors in the limits of are derived as expected.

Orthogonal polynomials and the finite Toda lattice
View Description Hide DescriptionThe choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter and the matrix representing multiplication by in terms of an orthogonal basis is constructed. It is then shown that the same induced time dependence for finitely supported distributions which gives the KP flow under the dual isomorphism induces the flow of the Toda hierarchy on the matrix. The corresponding solution is an particle, finite, nonperiodic Toda solution where is the cardinality of the support of plus the sum of the orders of the highest derivative taken at each point.

Period lengths of cellular automata with memory
View Description Hide DescriptionCellular automata have states 0 and 1, and their dynamics, driven by the local transition rule 90, can be simply represented with Laurent polynomials over a finite field . Cellular automata with memory, whose configurations are pairs of those of , are introduced as a useful machinery to solve certain equations on configurations, in particular, to compute fixed or kernel configurations of . This paper defines a notion of linear dynamical systems with memory, states their basic properties, and then studies some period lengths of onedimensional and twodimensional cellular automata with memory.

Advection diffusion past a strip. I. Normal incidence
View Description Hide DescriptionThe concentration p(x,y) of particles moving by diffusion and advection or drift is analyzed. The motion is impeded by an impenetrable strip that is parallel to the z axis. It is assumed that p satisfies the linear advectiondiffusion equation with a boundary condition on the strip. It is also assumed that p=1 at infinity. This problem is solved asymptotically for vL/D≫1, where v is the drift velocity, D is the diffusion coefficient, and 2L is the strip width. It is found that p is large on the side of the strip facing the incident flow, that p is small in the shadow behind the strip, and that p is nearly constant elsewhere. The case of a strip normal to the incident flow is rather different from that of a strip oblique to the flow. Methods of asymptotic analysis are used.

On solutions of constrained KP equations
View Description Hide DescriptionWe derive solutions of general Wronskian form for the (vector) constrained KP hierarchies. As one explicit example we discuss rational solutions. In order to introduce our method, we give a direct, elementary proof of the existence of Wronskian solutions for the lmodified KP hierarchies (l:0,1,…).

Exotic coherent structures in the (2+1) dimensional long dispersive wave equation
View Description Hide DescriptionIn this paper, we investigate the integrability aspects of the (2+1) dimensional coupled long dispersive wave (2LDW) equation introduced recently by Chakravarty, Kent, and Newman and establish its Painlevé (P) property. We then deduce its bilinear form from the P analysis and use it to construct wave type solutions for the field variables. We then identify line solitons for the composite field variable “qr” which eventually helps to bring out the peculiar localization behavior of the system by generating localized structures (dromions) for the composite field from out of only one ghost soliton driving the boundary. We have then extended this analysis to multidromion solutions.