Spectral Asymptotics of the Non-Self-Adjoint Harmonic Oscillator

An explicit asymptotic formula is obtained for the norms ofthe spectral projections of the non-self-adjoint harmonic oscillatorH. It is deduced that the spectral expansion of e^-Ht is normconvergent if and only if t is greater than a certain explicitpositive constant.     

Spectral Characterization of Some Cubic Graphs

It is proved that the Cartesian product of an odd cycle with the complete graph on 2 vertices, is determined by the spectrum of the adjacency matrix. We also present some computational results on the spectral characterization of cubic graphs on at most 20 vertices.     

Approximation of p-multiplier Operators via their Spectral Projections

Conditions for a p-multiplier $\psi: {\mathbb{Z}} \to {\mathbb{C}}Conditions for a p-multiplier are presented which ensure that the corresponding operator T_ψ, acting in , can be approximated by linear combinations of p-multiplier projections coming from the uniform operator closed, unital algebra of operators generated by T_ψ. Functions of bounded variation on play an important role, as do certain Λ (p)-sets. Dedicated to the memory of H. H. Schaefer Werner J. Ricker: Former Alexander von Humboldt Fellow at the Universit?t Tübingen, hosted by Prof. H.H. Schaefer from Sept. 1987 – Feb. 1988.     

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\) . Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\) , then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\) . The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.     

Spectral Asymptotics of the Harmonic Oscillator Perturbed by Bounded Potentials

Consider the operator in where q is a real function with q′ and bounded. The spectrum of T is purely discrete and consists of simple eigenvalues. We determine their asymptotics and we extend these results for complex q.Communicated by Bernard Helffersubmitted 23/04/04, accepted 26/10/04     

Approximation in the Space of Spectral Data of a Perturbed Harmonic Oscillator

We consider the spaces of sequences that arise in the inverse spectral problem for a harmonic oscillator perturbed by potential. We establish embedding theorems which connect such spaces with weight spaces . We also study the approximation of elements by finite sequences.This problem arises if we restore the potential from its spectrum.Bibliography: 5 titles.     

Proof of the Absence of Elliptic Solutions of the Cubic Complex Ginzburg-Landau Equation

We consider the cubic complex Ginzburg-Landau equation. Using Hone's method, based on formal Laurent-series solutions and the residue theorem, we prove the absence of elliptic standing-wave solutions of this equation. This result complements a result by Hone, who proved the nonexistence of elliptic traveling-wave solutions. We show that it is more efficient to apply Hone's method to a system of polynomial differential equations rather than to an equivalent differential equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 161–171, January, 2006.     

Error Bounds for Perturbation of the Drazin Inverse of Closed Operators with Equal Spectral Projections

We study perturbations of the Drazin inverse of a closed linear operator A for the case when the perturbed operator has the same spectral projection as A . This theory subsumes results recently obtained by Wei and Wang, Rako ) evi ' and Wei, and Castro and Koliha. We give explicit error estimates for the perturbation of Drazin inverse, and error estimates involving higher powers of the operators.     

Asymptotics of Spectral Data of a Harmonic Oscillator Perturbed by a Potential

Asymptotics of spectral data of a perturbed harmonic oscillator −y″ + x²y + q(x)y are obtained for potentials q(x) such that q′, xq ∈ L²(ℝ). These results are important in the solution of the corresponding inverse spectral problem. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 223–271.     

关于奇异积分三次复样条近似的误差阶估计

讨论曲线上柯西型奇异积分利用三次复样条进行近似计算的误差估计,对于相关函数类给出了这类逼近的误差阶.     

Simulation of Wave Propagation in Complex Structures with the Spectral Element Method

In nondestructive testing, the use of ultrasonic elastic waves has proven as one of the most successful principles to detect structural damage like cracks, delaminations etc. Especially, Structural Health Monitoring (SHM) is characterized by permanently installed embedded or surface–mounted actuators and sensors (e.g. piezoelectric patches). The capability of most approaches strongly depends on adequate choice of parameters like excitation signals and actuator/sensor positions. For this reason there is a growing interest in efficient and accurate simulation tools to shorten time and cost of the necessary pretests. With respect to high frequency excitation a computationally efficient method is required. This contribution presents the theoretical background of the spectral element method including the electro–mechanical coupling of piezo elements. The spectral element method generates a diagonal mass matrix leading to significant savings of memory and to a crucial reduction of complexity of the time integration algorithm. Both in–plane and out–of–plane waves can be handled. Numerical examples for the propagation of waves in stiffened structures are presented. The effect of improper placement of actuators/sensors is shown. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

度量投影的收敛性（英文）

讨论了赋范空间中度量投影的收敛性．得到了在局部紧集控制下，Chebyshev凸集序列的度量投影的收敛性与K－M收敛，Wijsman收敛和Kuratowski收敛都等价．本文的结论完善了M．Tsukada在[1]和[2]结果．     

On the Projections of Multivalued Maps

This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.     

Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials

We study the wave equation on one-dimensional self-similar fractal structures that can be analyzed by the spectral decimation method. We develop efficient numerical approximation techniques and also provide uniform estimates obtained by analytical methods.     

Charlier Polynomials and Charlier Oscillator as Discrete Realization of the Harmonic Oscillator

The Charlier polynomials are used for constructing oscillator-like systems. We introduce coherent states of the system are defined and study the main properties (in particular, the (over)completeness property). We show that the coherent states on the associated uncertainty relation take the minimum value. In the case under consideration, the Mandel parameter vanishes, which corresponds to the Poisson statistics of quasiexcitation spectrum for the considered oscillator. Bibliography: 30 titles.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 3–15.     

On the Norm of the Metric Projections

LetXbe a Banach space. GivenMa subspace ofXwe denote withP_Mthe metric projection ontoM. We defineπ(X) sup{P_M: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=L_p, we obtain the inequality P_M2^|2/p−1|, for every subspaceMofL_p. We also show thatπ(X)=π(X*).     

The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product

We use elementary methods to compute the L ²-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk and Zuk, and later used by them, together with Linnell and Schick, to produce a counterexample to a strong version of the Atiyah conjecture about the range of L ²-Betti numbers. We use our results to construct manifolds with certain L ²-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.