Fractal structure in the collision of vector solitons

We study the collision of two orthogonally polarized and equal-amplitude vector solitons in the nonintegrable coupled nonlinear Schrodinger equations. We show that the separation velocity versus collision velocity graph has a fractal structure. When we zoom into this graph, we get a structure qualitatively identical to the original one. In addition, collision dynamics in the zoomed-in windows is intimately related to that in the original graph. We explain this fractal dependence of the collision by a resonance mechanism between the translational motion of vector solitons and internal oscillations inside a vector soliton.     

Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

We consider a random Schr?dinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman–Molchanov method to prove localization. Received: 1 June 1998 / Accepted: 29 January 1999     

Anderson Localization at Band Edges for Random Magnetic Fields

We consider a magnetic Schrödinger operator in two dimensions. The magnetic field is given as the sum of a large and constant magnetic field and a random magnetic field. Moreover, we allow for an additional deterministic potential as well as a magnetic field which are both periodic. We show that the spectrum of this operator is contained in broadened bands around the Landau levels and that the edges of these bands consist of pure point spectrum with exponentially decaying eigenfunctions. The proof is based on a recent Wegner estimate obtained in Erd?s and Hasler (Commun. Math. Phys., preprint, arXiv:1012.5185) and a multiscale analysis.     

On the Dense Point and Absolutely Continuous Spectrum for Hamiltonians with Concentric <Emphasis Type="Italic">δ</Emphasis> Shells

We consider Schrödinger operators in dimension ν ≥ 2 with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum covers a half line determined by the appropriate one-dimensional comparison operator; it is dense pure point in the gaps of the latter. If the interaction is nontrivial and radially periodic, there are infinitely many absolutely continuous bands; in contrast to the regular case the lengths of the p.p. segments interlacing with the bands tend asymptotically to a positive constant in the high-energy limit.     

Magneto-metric Hamiltonians on quantum surfaces in the configuration space

The paper deals with the Schrödinger operator on a generic Riemann surface in a generic magnetic field. On a class of nano-momentum states, this operator is reduced to a system with one degree of freedom whose (quantized) phase space is the given configuration surface. The Hamiltonian of the reduced system is a joint geometric invariant of metric and magnetic tensors. Its spectrum can be computed semiclassically, which provides the asymptotics of eigenvalue clusters for the original Schrödinger operator near the ground level.     

A numerical study of the 2-flavour Schwinger model with dynamical overlap hypercube fermions

We present numerical results for the 2-flavour Schwinger model with dynamical chiral lattice fermions. We insert an approximately chiral hypercube Dirac operator into the overlap formula to construct the overlap hypercube operator. This is an exact solution to the Ginsparg?CWilson relation, with an excellent level of locality and scaling. Due to its similarity with the hypercubic kernel, a low polynomial in this kernel provides a numerically efficient Hybrid Monte Carlo force. We measure the microscopic Dirac spectrum and discuss the corresponding scale-invariant parameter, which takes a surprising form. This is an interesting case, since Random Matrix Theory is unexplored for this setting, where the chiral condensate ?? vanishes in the chiral limit. We also measure ?? and the ??pion?? mass, in distinct topological sectors. In this context we discuss and probe the topological summation of observables by various methods, as well as the evaluation of the topological susceptibility. The feasibility of this summation is essential for the prospects of dynamical overlap fermions in QCD.     

Spectra of Schrödinger Operators on Equilateral Quantum Graphs

We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum graph is the preimage of the combinatorial spectrum under a certain entire function. Using this Correspondence we show that the number of gaps in the spectrum of the Schrödinger operators admits an estimate from below in terms of the Hill operator independently of the graph structure.     

Exact Eigenfunctions of N-body System with Quadratic Pair Potential

We obtain the energy spectrum and all the corresponding eigenfunctions of N-body Bose and Fermi systems with Quadratic Pair Potentials in one dimension. The original first excited state or energy level is disappeared in one dimension, which results from the operation of symmetry or antisymmetry of identical particles. In two and higher dimensions, we give the energy spectrum and the analytical ground state wave functions and the degree of degeneracy. By comparison, we refine Avinash Khare's results by making some items in his article precisely.     

Exact Eigenfunctions of N-body System with Quadratic Pair Potential

We obtain the energy spectrum and all the corresponding eigenfunctions of N-body Bose and Fermi systems with Quadratic Pair Potentials in one dimension. The original first excited state or energy level is disappeared in one dimension, which results from the operation of symmetry or antisymmetry of identical particles. In two and higher dimensions, we give the energy spectrum and the analytical ground state wave functions and the degree of degeneracy. By comparison, we refine Avinash Khare's results by making some items in his article precisely.     

姜油细胞原位拉曼光谱研究

提出一种用拉曼光谱原位分析新鲜姜油细胞中姜油主成分的方法。用徒手切片制备新鲜姜样品,该样品置于DXR 激光共焦显微拉曼光谱仪下,用20倍物镜观察到油细胞,将激光聚焦在该油细胞上,获得了姜油细胞中姜油的拉曼光谱,共21条谱峰。不同油细胞上获得的拉曼光谱非常相似。获得了姜精油的拉曼光谱,与姜精油拉曼光谱的37条谱峰比较,油细胞有19条谱峰与之有对应关系。为了解释油细胞精油及姜精油的拉曼光谱,用密度泛函理论计算了姜烯的拉曼光谱。姜精油拉曼光谱有31条谱峰,油细胞中有19条谱峰与计算光谱有对应关系。该研究提供了一种拉曼光谱技术与密度泛函理论计算结合的快速容易的精油质量控制方法。     

Relative Oscillation Theory,Weighted Zeros of the Wronskian,and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm–Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein’s spectral shift function is established. Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.     

大高良姜与节鞭山姜油细胞原位拉曼光谱研究

为避免复杂的样品的制备及提取过程,最大限度避免精油活性成分变化,在常温下,用拉曼光谱原位分析大高良姜与节鞭山姜油细胞中精油。两种样品的拉曼光谱非常相似,较强峰出现在1636/1643、1605/1607、1291/1293、1199/1204、1168/1168cm~(-1),中等强度的峰出现在1585/-、1439/1437、1308/-、907/908、856/856、797/788、731/732、632/633cm~(-1)。说明两者的挥发油主要成分相同。与1’一乙酰氧基胡椒酚乙酸酯(1’-acetoxychavico lacetate)拉曼光谱主要的16条谱线比较,在节鞭山姜油细胞拉曼谱的18条谱峰中有15条与之对应,而对大高良姜而言19条谱线中,有14条与之对应。用密度泛函理论计算了ACA的拉曼光谱,并对谱线进行了初步的归属。     

基于小波分解的光谱特征提取算法研究

物质的光谱曲线反映了其独特的反射特性,利用光谱可以进行物质的分类识别工作。由于光谱曲线数据量较大、吸收特征不明显等特点,光谱曲线的特征提取是高光谱影像分类识别中的一个关键问题之一。该文利用小波分析技术通过对原始信号的分解,以及测量目标光谱特征的吸收宽度,确定了小波分解尺度,达到突出目标光谱吸收特征而抑制非相关特征及噪声的目的。通过实验表明,该方法可有效地降光谱维数,提高了光谱匹配识别精度。     

Topological Boundary Invariants for Floquet Systems and Quantum Walks

A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks.     

On the Spectral Behaviour of a Non-self-adjoint Operator with Complex Potential

We consider the non-self-adjoint Anderson operator with a complex potential as a pseudo-ergodic operator in one spatial dimension and use second order numerical ranges to obtain tight bounds on the spectrum of the operator. We also find estimates for the size of possible holes contained in the spectrum of such an operator.      

Quantum mechanics in discrete space and angular momentum

Recently we have studied quantum mechanics of bounded operators with a discrete spectrum. In particular, we derived an expression for the commutator[Q, P] of two bounded operators whose spectrum is discrete, and we showed that in the limit of a continuous spectrum the commutator becomes the standard one of Heisenberg. In this paper we show that the angular momentum operator and the phase operator satisfy the new commutation relation. We also briefly discuss the problem of the canonical phase operator conjugate to the number operator.     

Fluctuations in mean-field self-organized criticality

We present two models that exhibit self-organized criticality at the mean-field level. These can be variously interpreted in epidemiological or chemical reaction terms. By studying the master equation for these models we find, however, that only in one of them does the self-organized critical behavior survive in the face of fluctuations. For this model we show the spectrum of the evolution operator to have spectral collapse, i.e., instead of a gap, as would occur in noncritical behavior, there are eigenvalues that approach zero as an inverse power of system size.     

On Irreducible Second-Order Darboux Transformations

The second-order Darboux transform for a growing potential of a Schrödinger equation is discussed in detail. Restrictions on the transforming functions whose eigenvalues are greater than the energy of the ground state if the potential of the transformed Schrödinger equation is regular has been established. It has been shown that, along with the well-known elimination of two levels of the discrete spectrum, other opportunities can also be realized: (a) the transformed spectrum remains identical to the original one; (b) one level is eliminated from the spectrum; (c) one additional level is generated, and (d) two additional levels are generated.     

Water waves as a spatial dynamical system; infinite depth case

We review the mathematical results on traveling waves in one or several superposed layers of potential flow, subject to gravity, with or without surface and interfacial tension, where the bottom layer is infinitely deep. The problem is formulated as a "spatial dynamical system," and it is shown that the linearized operator of the resulting reversible system has an essential spectrum filling the real line. We consider three cases where bifurcation occurs. (i) The first case is when, in moving a parameter, two pairs of imaginary eigenvalues merge into one pair of double eigenvalues, and then split into four symmetric complex conjugate eigenvalues. (ii) The second case is when one pair of imaginary eigenvalues meet in 0, and disappear; (iii) the third case is when the phenomenon described in (ii) is superposed to the presence of another pair of imaginary eigenvalues sitting at finite distance from 0. We give a physical example for each case and more specially study the solitary waves and generalized solitary waves, emphasizing the differences, in the methods and in the results, between these cases and the finite depth case.