An Index Theory For Quantum Dynamical Semigroups

W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

     

Accuracy of semiclassical dynamics in the presence of chaos

We review some of the issues facing semiclassical methods in classically chaotic systems, then demonstrate the long-time accuracy of semiclassical propagation of a nonstationary wave packet using the quantum baker's map of Balazs and Voros. We show why some of the standard arguments against the efficacy of semiclassical dynamics for long-time chaotic motion are incorrect.     

A variational problem with lack of compactness for H(S;S) maps of prescribed degree

We consider, for maps in H^1/2(S¹;S¹), a family of (semi)norms equivalent to the standard one. We ask whether, for such a norm, there is some map in H^1/2(S¹;S¹) of prescribed topological degree equal to 1 and minimal norm. In general, the answer is no, due to concentration phenomena. The existence of a minimal map is sensitive to small perturbations of the norm. We derive a sufficient condition for the existence of minimal maps. In particular, we prove that, for every given norm, there are arbitrarily small perturbations of it for which the minimum is attained. In case there is no minimizer, we determine the asymptotic behavior of minimizing sequences. We prove that, for such minimizing sequences, the energy concentrates near a point of S¹. We describe this concentration in terms of bubbling-off of circles.     

An Estimate of Smoothing and Composition with Applications to Conjugation Problems

An estimation of the difference between smoothing a composition of two functions and composing their smoothings is given. The smoothing operator considered here is defined by the convolution operator with a holomorphic kernel. It is also shown, by means of an example, how to apply the estimate given here to obtain finite differentiable versions of theorems on conjugation of maps in non-perturbative settings.      

Graph Approach to the Computation of the Homology of Continuous Maps

We introduce an efficient algorithm to compute the homomorphism induced in (relative) homology by a continous map. The algorithm is based on a cubical approximation of the map and the theory of multivalued maps. A software implementation of the algorithms introduced in this paper is available at [27].     

Iterations of random and deterministic functions

Letf be a probability generating function on [0, 1]. The convergence of its iteratesf _n to fixed points is studied in this paper. Results include rates forf andf ^-1. Also iterates of independent identically distributed stable processes are studied and a trichotomy based on the order of the stability is established. Dedicated to the memory of Professor K G Ramanathan     

Self-dual connections,hyperbolic metrics and harmonic mappings on Riemann surfaces

The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.     

A finiteness theorem of harmonic maps from compact lie groups toO HS(H)

In this article we study the behavior of harmonic maps from compact connected Lie groups with bi-invariant metrics into a Hilbert orthogonal group. In particular, we will demonstrate that any such harmonic map always has image contained within someO(n),n<∞. Since homomorphisms are a special subset of the harmonic maps we get as a corollary an extension of the Peter-Weyl theorem, namely, that every representation of a connected compact Lie group is finite dimensional. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Density-difference maps in quantum chemistry

Electron density-difference maps, used to study the changes that occur when a molecule changes its state or when the nuclei of a molecule change their relative positions, are generally useful only if the atomic densities cancel when one molecular density distribution is subtracted from the other. When, as in the case of the nonrigid internal rotation in ethane, such a cancellation of atomic densities is not possible the method of simple subtraction is no longer appropriate. It is shown that useful density-difference maps can nevertheless be obtained, when the changes in geometric parameters are small, by the calculation of two generalized density-difference functions: a point difference function which allows a comparison of the densities at corresponding points in the two systems, and a volume difference function to compare the amounts of charge in corresponding regions. The method is illustrated by consideration of a change in bond length of the nitrogen molecule and by the nonrigid internal rotation in ethane.Presented at the XIVth Quantum Theory Conference in Canterbury, England, September 1981.