m-可乘序列谱测度的关联维数

作者利用关联函数的递减速度与Fourier谱特征之间的关系,计算出无穷m -可乘序列谱测度的关联维数. 且通过对m -可乘序列关联函数的研究,验证了其谱测度是奇异连续的结论.     

Critical fluctuation in daily incidence of acute myocardial infarction

Continuous periodogram power spectral analysis of daily incidence of acute myocardial infarction (AMI) reported at a hospital for cardiology in Pune, India for the two-year period June 1992–May 1994 show that the power spectra follow the universal and unique inverse power law form of the statistical normal distribution. The same time inverse power law form for power spectra of space-time fluctuations are also ubiquitous to dynamical systems in nature and have been identified as signatures of self-organized criticality. The unique quantification for self-organized criticality presented in this paper is shown to be intrinsic to quantumlike mechanics governing fractal space-time fluctuation patterns in dynamical systems and suggest a possibly fruitful relation and analogy between different subject such as chaos, diffusion and quantum physics. The results found which mimic those obtained in quantum physics by El Naschie using the concept of Cantorian space ε^(∞) suggest that, that tools developed in some of these areas may be used advantageously in the medical field as pioneered by A.T. Winfree [Int. J. Bifurcation and Chaos 7 (3) (1997) 487–526] and A.V. Holden [Chaos, Solitons and Fractals 5 (3/4) (1995) 691–704; Int. J. Bifurcation and Chaos 7 (9) (1997) 2075–2104].     

Spectral Dimension of Liouville Quantum Gravity

This paper is concerned with computing the spectral dimension of (critical) 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as derived by physicists (see Ambjørn et al. in JHEP 9802:010, 1998) 15 years ago.     

Twisted spectral triples and quantum statistical mechanical systems

Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number fields, spin manifolds, graphs. There are similarities between the two structures, and we show that the notion of twisted spectral triple, introduced recently by Connes and Moscovici, provides a natural bridge between them. We investigate explicit examples, related to the Bost-Connes quantum statistical mechanical system and to Riemann surfaces and graphs.     

Two-time correlation functions in an exactly solvable spin-boson model

We consider a spin-boson model describing the dephasing process in an open quantum system and obtain exact expressions for the two-time spin correlation function and the decoherence function applicable for any values of the coupling constants. We show that the initial statistical correlations between the dynamical system and the heat bath considerably affect the time dependence of the decoherence function.     

Memory Effects and Nonequilibrium Correlations in the Dynamics of Open Quantum Systems

We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.     

Lie algebras and recurrence relations I

We study representations of the Heisenberg-Weyl algebra and a variety of Lie algebras, e.g., su(2), related through various aspects of the spectral theory of self-adjoint operators, the theory of orthogonal polynomials, and basic quantum theory. The approach taken here enables extensions from the one-variable case to be made in a natural manner. Extensions to certain infinite-dimensional Lie algebras (continuous tensor products, q-analogs) can be found as well. Particularly, we discuss the relationship between generating functions and representations of Lie algebras, spectral theory for operators that lead to systems of orthogonal polynomials and, importantly, the precise connection between the representation theory of Lie algebras and classical probability distributions is presented via the notions of quantum probability theory. Coincidentally, our theory is closed connected to the study of exponential families with quadratic variance in statistical theory.     

Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm

We obtain results on small deviations of Bogoliubov’s Gaussian measure occurring in the theory of the statistical equilibrium of quantum systems. For some random processes related to Bogoliubov processes, we find the exact asymptotic probability of their small deviations with respect to a Hilbert norm.     

Statistical mechanics of relaxation processes in alternating fields

Using the nonequilibrium statistical operator method and the projection technique, we derive the system of exact relaxation equations for a quantum system interacting with an alternating external field. These equations hold in the case where some of the basic dynamical variables describing a nonequilibrium state depend explicitly on time. We obtain the exact expression for the entropy production in an alternating external field. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 102–112, January, 2008.     

Exact <Emphasis Type="Italic">S</Emphasis>-wave solution of the Schrödinger equation for three new potentials using the transformation method

We apply the extended transformation method to a non-power-law potential to generate a set of exactly solvable quantum systems in spaces of any dimensions. We derive exact analytic solutions of the Schrödinger equations with an exactly solvable non-power-law potential. For the transformed potentials obtained as a result, we calculate the quantized bound-state energy spectra and the corresponding wave functions.     

Gauge theories as matrix models

We discuss the relation between the Seiberg-Witten prepotentials, Nekrasov functions, and matrix models. On the semiclassical level, we show that the matrix models of Eguchi-Yang type are described by instantonic contributions to the deformed partition functions of supersymmetric gauge theories. We study the constructed explicit exact solution of the four-dimensional conformal theory in detail and also discuss some aspects of its relation to the recently proposed logarithmic beta-ensembles. We also consider “quantizing” this picture in terms of two-dimensional conformal theory with extended symmetry and stress its difference from the well-known picture of the perturbative expansion in matrix models. Instead, the representation of Nekrasov functions using conformal blocks or Whittaker vectors provides a nontrivial relation to Teichmüller spaces and quantum integrable systems.     

Quantum Dot and Antidot Infrared Photodetectors: Iterative Methods for Solving the Laplace Equation in Domains with Involved Geometry

We propose iteration methods for solving the Dirichlet problem in domains with involved geometry. Such problems arise in relation to the problem of optimizing quantum dot and antidot infrared detectors. We estimate the deviation of an approximate solution from the exact solution.     

Schrödinger uncertainty relation for a quantum oscillator in a thermostat

We introduce the Schrödinger correlator as the holistic characteristic of two types of fluctuation correlations in quantum dynamics and in statistical thermodynamics. We are the first to derive it using methods of thermofield dynamics for the coordinate-momentum variables of a quantum oscillator in a thermostat. We show that the obtained value ensures that the Schrödinger uncertainty relation becomes an equality at all temperatures. We find that the thermal equilibrium for the quantum oscillator has the sense of the thermal correlated coherent state and can be adequately described by a wave function with temperature-dependent amplitude and phase.     

Spectral inclusions for operators on Banach spaces

This article centers around the relation between the spectra of two Banach space operators that are linked by some intertwining condition such as quasi-similarity. Certain conditions from local spectral theory are shown to be both necessary and sufficient for these operators to have equal spectra, approximate point spectra, or surjectivity spectra. A key role is played by a localized version of Bishop’s classical property (β) and a related closed range condition. As an application to harmonic analysis, the measures on a locally compact abelian group that avoid the Wiener-Pitt phenomenon are characterized in terms of local spectral theory.     

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

We consider the nonlinear Schrödinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrödinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.     

The quantum drift-diffusion model: Existence and exponential convergence to the equilibrium

This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. in [7]. The model is obtained as the diffusive limit of the quantum Liouville–BGK equation, where the collision term is defined after a local quantum statistical equilibrium. The corner stone of the model is the closure relation between the density and the current, which is nonlinear and nonlocal, and is the main source of the mathematical difficulties. The question of the existence of solutions has been open since the derivation of the model, and we provide here a first result in a one-dimensional periodic setting. The proof is based on an approximation argument, and exploits some properties of the minimizers of an appropriate quantum free energy. We investigate as well the long time behavior, and show that the solutions converge exponentially fast to the equilibrium. This is done by deriving a non-commutative logarithmic Sobolev inequality for the local quantum statistical equilibrium.     

The functional integral in the Hubbard model

In a new functional integral approach proposed for the model, we find the regime with a deformed integration measure in which the standard integral is replaced with the Jackson integral. We indicate the relation to a p-adic functional integral. For the magnetic and electronic subsystems in the effective functional that results from the operator formulation of the Hubbard model, we find the two-parametric quantum derivative resulting in the appearance of the quantum SU_rq (2) group. We establish the relation to the one-parametric quantum derivative and to the standard derivative.     

Lax Pair for the Free-Fermion Eight-Vertex Model

An explicit result on the quantum Lax pair for the eight-vertex free-fermion model with external fields in both directions is exhibited. Its relation to the quantum spectral transform and its corresponding Hamiltonian—the XY model with a transverse field—is discussed. The Yang-Baxter relation is computed, and the equations of motion are solved.