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

The accuracy of merging approximation in generalized St. Petersburg games

Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games, extending the short expansions due to Cs\"org\H{o}, S., Merging asymptotic expansions in generalized St. Petersburg games, \textit{Acta Sci. Math. (Szeged)} \textbf{73} 297--331, 2007. These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions. The length of the expansion depends upon the tail parameter. Both uniform and nonuniform bounds are presented.Comment: 30 pages long version (to appear in Journal of Theoretical Probability); some corrected typo

A quantitative central limit theorem for linear statistics of random matrix eigenvalues

It is known that the fluctuations of suitable linear statistics of Haar distributed elements of the compact classical groups satisfy a central limit theorem. We show that if the corresponding test functions are sufficiently smooth, a rate of convergence of order almost $1/n$ can be obtained using a quantitative multivariate CLT for traces of powers that was recently proven using Stein's method of exchangeable pairs.Comment: Title modified; main result stated under slightly weaker conditions; accepted for publication in the Journal of Theoretical Probabilit

The singular continuous diffraction measure of the Thue-Morse chain

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product.Comment: 6 pages, 1 figure; revised and improved versio

The averaged null energy condition and difference inequalities in quantum field theory

Recently, Larry Ford and Tom Roman have discovered that in a flat cylindrical space, although the stress-energy tensor itself fails to satisfy the averaged null energy condition (ANEC) along the (non-achronal) null geodesics, when the ``Casimir-vacuum" contribution is subtracted from the stress-energy the resulting tensor does satisfy the ANEC inequality. Ford and Roman name this class of constraints on the quantum stress-energy tensor ``difference inequalities." Here I give a proof of the difference inequality for a minimally coupled massless scalar field in an arbitrary two-dimensional spacetime, using the same techniques as those we relied on to prove ANEC in an earlier paper with Robert Wald. I begin with an overview of averaged energy conditions in quantum field theory.Comment: 20 page

Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials

We study discrete quasiperiodic Schr\"odinger operators on \ell^2(\zee) with potentials defined by $\gamma$-H\"older functions. We prove a general statement that for $\gamma >1/2$ and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.Comment: 15 page

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

Analysis of Fourier transform valuation formulas and applications

The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in L\'evy and stochastic volatility models.Comment: 26 pages, 3 figures, to appear in Appl. Math. Financ

On infinite-volume mixing

In the context of the long-standing issue of mixing in infinite ergodic theory, we introduce the idea of mixing for observables possessing an infinite-volume average. The idea is borrowed from statistical mechanics and appears to be relevant, at least for extended systems with a direct physical interpretation. We discuss the pros and cons of a few mathematical definitions that can be devised, testing them on a prototypical class of infinite measure-preserving dynamical systems, namely, the random walks.Comment: 34 pages, final version accepted by Communications in Mathematical Physics (some changes in Sect. 3 -- Prop. 3.1 in previous version was partially incorrect