We study the presence of a logarithmic time scale in discrete approximations of Sawtooth Maps on the 2--torus. The techniques used are suggested by quantum mechanical similarities, and are based on a particular class of states on the torus, that fulfill dynamical localization properties typical of quantum Coherent States.
We construct concrete examples of time operators for both continuous and discrete-time homogeneous quantum walks, and we determine their deficiency indices and spectra. For a discrete-time quantum walk, the time operator can be self-adjoint if the ti
me evolution operator has a non-zero winding number. In this case, its spectrum becomes a discrete set of real numbers.
The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng [Z. B. Wu and J. Y. Zeng, Phys. Rev. A 62,032509 (2000)]. We find the similar properties in the responding
systems in a spherical space, whose dynamical symmetries are described by Higgs Algebra. There exists a conserved aphelion and perihelion vector, which, together with angular momentum, constitute the generators of the geometrical symmetry group at the aphelia and perihelia points $(dot{r}=0)$.
We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V_{sigma}(x) = -|V_{0}| |x|^{-sigma}, 0 < sigma leq 2. For these potentials the quasiclassical approximation for n -> infty predicts quantized energy le
vels e_{sigma}(n) of a bounded spectrum varying as e_{sigma}(n) ~ -n^{-2sigma/(2-sigma)}. We construct collective quantum states using the set of wavefunctions of the discrete spectrum taking into account this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For sigma in the range 0<sigmaleq 2/3 we present exact implementations of such states for the parametrization sigma = 2(k-l)/(3k-l), with k and l positive integers satisfying k>l.
This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs. In the quantum setting, the right leg of the pair identifies the Ber
ry curvature, while its left leg is shown to lead to more general realizations of the density operator which have recently appeared in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmanns density matrix and Koopman-von Neumann wavefunctions are shown to be special cases of this construction.
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable p
artial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.
Fabio Benatti
,Valerio Cappellini (Dipartimento di Fisica Teorica
,n Universita` di Trieste
.
(2004)
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"Continuous Limit of Discrete Sawtooth Maps and its Algebraic Framework"
.
Valerio Cappellini Dr.
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