Fundamental Limits on the Speed of Evolution of Quantum States

This paper reports on some new inequalities of Margolus-Levitin-Mandelstam-Tamm-type involving the speed of quantum evolution between two orthogonal pure states. The clear determinant of the qualitative behavior of this time scale is the statistics of the energy spectrum. An often-overlooked correspondence between the real-time behavior of a quantum system and the statistical mechanics of a transformed (imaginary-time) thermodynamic system appears promising as a source of qualitative insights into the quantum dynamics.Comment: 6 pages, 1 eps figur

Convergence and Stability of the Inverse Scattering Series for Diffuse Waves

We analyze the inverse scattering series for diffuse waves in random media. In previous work the inverse series was used to develop fast, direct image reconstruction algorithms in optical tomography. Here we characterize the convergence, stability and approximation error of the serie

Rigidity and Non-recurrence along Sequences

Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences. A variety of classes of sequences with various properties are considered showing the complicated and rich structure of rigid and non-recurrent sequences

Selfsimilarity and growth in Birkhoff sums for the golden rotation

We study Birkhoff sums S(k,a) = g(a)+g(2a)+...+g(ka) at the golden mean rotation number a with periodic continued fraction approximations p(n)/q(n), where g(x) = log(2-2 cos(2 pi x). The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena. We relate the boundedness of log- averaged Birkhoff sums S(k,a)/log(k) and the convergence of S(q(n),a) with the existence of an experimentally established limit function f(x) = lim S([x q(n)])(p(n+1)/q(n+1))-S([x q(n)])(p(n)/q(n)) for n to infinity on the interval [0,1]. The function f satisfies a functional equation f(ax) + (1-a) f(x)= b(x) with a monotone function b. The limit lim S(q(n),a) for n going to infinity can be expressed in terms of the function f.Comment: 14 pages, 8 figure

On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of \Co^{N}. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in \Co^{2}.Comment: minor changes in the formulation and the proof of Lemma 8.

Theory of Circle Maps and the Problem of One-Dimensional Optical Resonator with a Periodically Moving Wall

We consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions. Notably, well-known results in the theory of circle maps (which we review briefly) imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially. Even if these intervals are dense for typical motions of the reflecting boundary, in the complement there is a positive measure set of parameters where the energy remains bounded.Comment: 34 pages LaTeX (revtex) with eps figures, PACS: 02.30.Jr, 42.15.-i, 42.60.Da, 42.65.Y

A Bishop-Phelps-Bollobas Type Theorem for uniform algegras

This paper is devoted to showing that Asplund operators with range in a uniform Banach algebra have the Bishop¿Phelps¿Bollobas property, i.e., they are approximated by norm attaining Asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it. To prove this result we use the weak*-to-norm fragmentability of weak*-compact subsets of the dual of Asplund spaces and we need to observe a Urysohn type result producing peak complex-valued functions in uniform algebras that are small outside a given open set and whose image is inside a Stolz region.This research was partially supported by MEC and FEDER projects MTM2008-05396 and MTM2011-25377. The research of the second author was also partially supported by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project PAID-06-09-2829).Cascales, B.; Guirao Sánchez, AJ.; Kadets, V. (2013). A Bishop-Phelps-Bollobas Type Theorem for uniform algegras. Advances in Mathematics. 240:370-382. https://doi.org/10.1016/j.aim.2013.03.005S37038224

Time Asymmetric Quantum Physics

Mathematical and phenomenological arguments in favor of asymmetric time evolution of micro-physical states are presented.Comment: Tex file with 2 figure

Spectral and topological properties of a family of generalised Thue-Morse sequences

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is know as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.Comment: Dedicated to Robert V. Moody on the occasion of his 70th birthday; revised and improved versio