Zeta Function Resummation of Spectral Functions

This article continues an extensive analysis of spectral functions such as for both general and explicitly known spectra {λ_m}. In physical applications (which in quantum field theory are numerous) the spectral functions are mode sums. Our main analytic tool is the ζ-function resummation method which expresses f(s|x) in powers of x and perhaps other simple functions of x. Here the general spectrum will be replaced by its asymptotic form λ_m = (const) m^α with α > 0 (Weyl's theorem). This preserves certain global features of the general spectrum problem but enables one to work entirely in terms of known functions. This simplified problem will be fully analysed and certain aspects of it (in particular the continuum limit) studied for the first time. Several mode-sum calculations illustrate physical application of the method.     

Spectral Determinant of the Two-Photon Quantum Rabi Model

The various generalized spectral determinants (G-functions) of the two-photon quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only the G-function proposed by Chen et al. in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. This function is derived rigorously employing the ${\mathbb{Z}}_4$-symmetry of the model and shown that its zeros correspond to the complete regular spectrum.     

Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

We investigate the existence of the meromorphic extension of the spectral zeta function of a Laplacian on self-similar fractals using the results of Kigami and Lapidus (based on renewal theory) and the newer results by Hambly and Kajino based on heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true for the examples that have been analyzed in the existing literature.     

An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an ε-neighborhood of a given number λ > 0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.     

On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator

The spectral zeta function for the so-called noncommutative harmonic oscillator is able to be meromorphically extended to the whole complex plane, having only one simple pole at the same point s = 1 where Riemann's zeta function ζ(s) has, and possesses a trivial zero at each nonpositive even integer. The essential part of its proof is sketched. A new result is also given on the lower and upper bounds of the eigenvalues of the noncommutative harmonic oscillator.     

Spectral Element Discretization of the Stokes Equations in Deformed Axisymmetric Geometries

In this paper, we study the numerical solution of the Stokes system in deformed axisymmetric geometries. In the azimuthal direction the discretization is carried out by using truncated Fourier series, thus reducing the dimension of the problem. The resulting two-dimensional problems are discretized using the spectral element method which is based on the variational formulation in primitive variables. The meridian domain is subdivided into elements, in each of which the solution is approximated by truncated polynomial series. The results of numerical experiments for several geometries are presented.     

基于光谱成像与光谱分析的壁画鉴赏研究

针对壁画鉴赏中对科学分析方法的需求,根据光谱成像及光谱分析方法在壁画处理中的特点与优势,重点讨论了光谱成像技术在壁画鉴赏处理中的应用,其可以辅助传统的专家经验鉴定方法,提供更丰富客观的画作信息,实现作品的鉴赏分析。光谱成像与分析方法是一类重要的科学分析方法,因其具有非接触、破损少甚至无破损、科学、准确、可量化等特点,可应用于壁画修复、信息提取、颜料分析、特征增强、鉴赏分析等。首先,从技术实现角度出发给出具体的鉴赏应用方向。然后,基于光谱特征分析、主成分分析、独立分量分析、相关性判别、人机交互处理等方法,对实际观测的韩休墓壁画进行实验处理,验证技术应用的实验效果,最后,结合应用实验提出包括规范化采集、图谱特征分析、图谱处理系统等关键技术还需进一步解决。实验结果表明,利用光谱成像分析技术可以实现壁画颜料信息的判读,隐藏信息的提取、弱信息的增强,不同属性的分类识别,同时,还可以综合特殊需求,开展画作图谱探索性研究,如壁画处理中的涂改小孩分析、画作修复等。     

Zeta functions and the Casimir energy

We use zeta function techniques to give a finite definition for the Casimir energy of an arbitrary ultrastatic spacetime with or without boundaries. We find that the Casimir energy is intimately related to, but not identical to, the one-loop effective energy. We show that in general the Casimir energy depends on a normalization scale. This phenomenon has relevance to applications of the Casimir energy in bag models of QCD.Within the framework of Kaluza-Klein theories we discuss the one-loop corrections to the induced cosmological and Newton constants in terms of a Casimir like effect. We can calculate the dependence of these constants on the radius of the compact dimensions, without having to resort to detailed calculations.     

《光谱学与光谱分析》

<正>《光谱学与光谱分析》是中国光学学会主办的学术刊物。ISSN 1000-0593,CN 11-2200/O4;月刊,大16开本,288页。系中国物理类、化学类核心期刊;中国科协择优支持基础性、高科技学术期刊;已被国内外CSCD,SCI,Ei,CA,AA,PЖ,MEDLINE等     

激光微区发射光谱谱线品质的研究

利用单脉冲YAG激光、YGJ Ⅱ激光微区分析仪结合光电检测系统 ,在减压氩气环境下实验研究了金属分析样品发射光谱中CuⅠ 32 4 7nm和CuⅠ 32 7 4nm的时间特性、谱线强度及空间分布 ,并与空气环境下的实验结果进行了比较。实验结果表明在减压氩气环境下 ,激光微区发射光谱与环境气体、环境气压和辅助激发参数密切相关。当辅助电极端面直径为 1 5mm、辅助电极距分析样品表面高度为 4mm、辅助电极间距为3mm、辅助激发电压为 1 30 0V、氩气压力 33 2kPa时 ,CuⅠ 32 4 7nm和CuⅠ 32 7 4nm谱线的发射时间比空气下延长了 50 0 μs,谱线强度约为相同气压空气环境下的 4倍 ,约为 1个大气压空气环境下谱线强度的 2倍 ,谱线的半最大值宽度明显变窄。因此氩气环境延长了谱线的发射时间 ,减少了自吸效应 ,使谱线强度明显增强 ,谱线品质得到显著改善。     

Periodicity and the Determinant Bundle

The infinite matrix ‘Schwartz’ group G ^−∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G ^−∞. We show that while the higher (even, Schwartz) loop groups of G ^−∞, again classifying for odd K-theory, do not carry multiplicative determinants generating the first Chern class, ‘dressed’ extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding τ invariant is a determinant in this sense. The first author acknowledges the support of the National Science Foundation under grant DMS0408993, the second author acknowledges support of the Fonds québécois sur la nature et les technologies and NSERC while part of this work was conducted.     

Determinant Bundles, Manifolds with Boundary and Surgery¶II. Spectral Sections and Surgery Rules for Anomalies

Let be a closed fibration of Riemannian manifolds and let , be a family of generalized Dirac operators. Let be an embedded hypersurface fibering over B; . Let be the Dirac family induced on . Each fiber in is the union along of two manifolds with boundary . In this paper, generalizing our previous work[16], we prove general surgery rules for the local and global anomalies of the Bismut–Freed connection on the determinant bundle associated to . Our results depend heavily on the b-calculus [12], on the surgery calculus [11] and on the APS family index theory developed in [13], in particular on the notion of spectral section for the family . Received: 23 October 1996 / Accepted: 28 July 1997     

铜绿颜料的光谱分析及稳定性研究

在中国颜料使用历史中,铜绿是绿色颜料中使用较为普遍的一种,具有使用地域广、时间长的特点。以碱式氯化铜为主要成分的铜绿为研究对象,探讨了其性质、四种同分异构体的光谱特性及热力学稳定性,并介绍了在彩绘文物中的应用。结果表明,采用拉曼光谱分析可以快速鉴别铜绿四种同分异构体,且对样品仅微损甚至无损。碱式氯化铜四种同分异构体的稳定性大小依次为：斜氯铜矿＞副氯铜矿＞氯铜矿＞羟氯铜矿。对文物样品中的铜绿进行鉴别发现样品中的铜绿多为羟氯铜矿和氯铜矿,属于四种同分异构体中较不稳定的状态。根据奥斯特瓦尔德规则(Ostwald step rule)的相关理论,建议在今后的保护工作中要加强对所属彩绘文物环境等的监控力度,以防止引起彩绘文物发生物理、化学结构的改变。     

Zeta Function on a Generalised Cone

The analytic properties of the -function for a Laplace operator on a generalised cone are studied in some detail using Cheeger's approach and explicit expressions are given. In the compact case, the -function of the Laplace operator turns out to be singular at the origin. As a result, strictly speaking, the -function regularisation does not regularise and a further subtraction is required for the related one-loop effective potential.     

基于矩阵分析的STR荧光解谱技术

用多种荧光标记物进行STR检测时,由于荧光光谱的谱带展宽特性,各个荧光光谱之间有重叠部分,如何实现将相互重叠的各个波长上的能量有效利用起来,对提高荧光利用效率至关重要。本文给出了一种基于矩阵分析的数据处理方法,该方法在进行荧光探测前首先要对荧光谱进行光谱校正,即要得到每种所用标记染料受激发射荧光的光谱分布,根据光谱分布建立染料组合荧光信号矩阵,然后对矩阵进行归一,利用归一后的矩阵对所探测的荧光谱进行解谱,从而得到所期望的荧光谱图。理论分析和实验结果表明,该方法可以有效的利用各个波长的荧光光谱能量,并实现对不同荧光重叠谱的有效光谱解谱。     

Hierarchy of the Selberg Zeta Functions

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.Mathematics Subject Classifications (2000). Primary 11M36, Secondary 33B15     

Analytic Surgery of the Zeta Function

In this paper we study the asymptotic behavior (in the sense of meromorphic functions) of the zeta function of a Laplace-type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface, separating the manifold into two manifolds with infinite cylindrical ends. We also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary, forming a manifold with an infinite cylindrical end. Such singular deformations fall under the category of “analytic surgery”, developed originally by Hassell (Comm Anal Geom 6:255–289, 1998), Hassell et al. (Comm Anal Geom 3:115–222, 1995) and Mazzeo and Melrose (Geom Funct Anal 5:14–75, 1995) in the context of eta invariants and determinants.     

Statistical Analysis of Simulated Random Packings of Spheres

This paper describes two algorithms for the generation of random packings of spheres with arbitrary diameter distribution. The first algorithm is the force‐biased algorithm of Mościński and Bargieł. It produces isotropic packings of very high density. The second algorithm is the Jodrey‐Tory sedimentation algorithm, which simulates successive packing of a container with spheres following gravitation. It yields packings of a lower density and of weak anisotropy. The results obtained with these algorithms for the cases of log‐normal and two‐point sphere diameter distributions are analysed statistically, i. e. standard characteristics of spatial statistics such as porosity (or volume fraction), pair correlation function of the system of sphere centres and spherical contact distribution function of the set‐theoretical union of all spheres are determined. Furthermore, the mean coordination numbers are analysed. These results are compared for both algorithms and with data from the literature based on other numerical simulations or from experiments with real spheres.