Action-Angle variables for the Gel'fand-Dikii flows

Using the scattering transform for $n^{th}$ order linear scalar operators, the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner Poisson bracket for the KdV hierarchy, is computed on the scattering side. Action-angle variables are then constructed. Using this, complete integrability is demonstrated in the strong sense. Real action-angle variables are constructed in the self-adjoint case

The geometric sense of R. Sasaki connection

For the Riemannian manifold $M^{n}$ two special connections on the sum of the tangent bundle $TM^{n}$ and the trivial one-dimensional bundle are constructed. These connections are flat if and only if the space $M^{n}$ has a constant sectional curvature $\pm 1$. The geometric explanation of this property is given. This construction gives a coordinate free many-dimensional generalization of the connection from the paper: R. Sasaki 1979 Soliton equations and pseudospherical surfaces, Nuclear Phys., {\bf 154 B}, pp. 343-357. It is shown that these connections are in close relation with the imbedding of $M^{n}$ into Euclidean or pseudoeuclidean $(n+1)$-dimension spaces.Comment: 7 pages, the key reference to the paper of Min-Oo is included in the second versio

The Optimal Single Copy Measurement for the Hidden Subgroup Problem

The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup problem. Previous work has identified the optimal single copy measurement for the hidden subgroup problem over abelian groups as well as for the non-abelian problem in the setting where the subgroups are restricted to be all conjugate to each other. Here we describe the optimal single copy measurement for the hidden subgroup problem when all of the subgroups of the group are given with equal a priori probability. The optimal measurement is seen to be a hybrid of the two previously discovered single copy optimal measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe

Neumark Operators and Sharp Reconstructions, the finite dimensional case

A commutative POV measure $F$ with real spectrum is characterized by the existence of a PV measure $E$ (the sharp reconstruction of $F$) with real spectrum such that $F$ can be interpreted as a randomization of $E$. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark's extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of $F$ and the sharp reconstruction of $F$. The relevance of this result to the theory of non-ideal quantum measurement and to the definition of unsharpness is analyzed.Comment: 37 page

Wave Solutions of Evolution Equations and Hamiltonian Flows on Nonlinear Subvarieties of Generalized Jacobians

The algebraic-geometric approach is extended to study solutions of N-component systems associated with the energy dependent Schrodinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvariaties of Jacobi varieties. The systems under study include the shallow water equation and Dym type equation. The classes of solutions are described in terms of theta-functions and their singular limits by using new parameterizations. A qualitative description of real valued solutions is provided

Effects of Gravitational Microlensing on P-Cygni Profiles of Type Ia Supernovae

A brief description of the deformed spectra of microlensed SNe Ia is presented. We show that microlensing amplification can have significant effects on line profiles. The resonance-scattering code SYNOW is used to compute the intensity profile in the rest frame of the supernova. The observed (microlensed) spectral lines are predicted assuming a simple stellar-size deflector, and are compared to unlensed cases to show the effects microlensing by solar-size deflectors can have on spectral lines. We limit our work to spherically symmetric deflectors.Comment: 18 pages, 9 figures, references added, submitted to Ap

Communication Capacity of Quantum Computation

By considering quantum computation as a communication process, we relate its efficiency to a communication capacity. This formalism allows us to rederive lower bounds on the complexity of search algorithms. It also enables us to link the mixedness of a quantum computer to its efficiency. We discuss the implications of our results for quantum measurement.Comment: 4 pages, revte

Adversary lower bounds for nonadaptive quantum algorithms

International audienceWe present general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Our results are based on the adversary method of Ambainis

New summing algorithm using ensemble computing

We propose an ensemble algorithm, which provides a new approach for evaluating and summing up a set of function samples. The proposed algorithm is not a quantum algorithm, insofar it does not involve quantum entanglement. The query complexity of the algorithm depends only on the scaling of the measurement sensitivity with the number of distinct spin sub-ensembles. From a practical point of view, the proposed algorithm may result in an exponential speedup, compared to known quantum and classical summing algorithms. However in general, this advantage exists only if the total number of function samples is below a threshold value which depends on the measurement sensitivity.Comment: 13 pages, 0 figures, VIth International Conference on Quantum Communication, Measurement and Computing (Boston, 2002

Inverse Scattering Transform for the Camassa-Holm equation

An Inverse Scattering Method is developed for the Camassa-Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are explicitly constructed in terms of the scattering data. The main difference with respect to the standard Inverse Scattering Transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.Comment: 17 pages, LaTe