Perturbative and nonperturbative parts of eigenstates and local spectral density of states: the wigner-band random-matrix model

The Wigner-band random-matrix model is studied by making use of a generalization of Brillouin-Wigner perturbation theory. Energy eigenfunctions are shown to be divided into perturbative and nonperturbative parts. Several perturbation strengths predicted by the perturbation theory are found to play important roles in the variation of the shape of the local spectral density of states with perturbation strength.     

Interplay between edge states and simple bulk defects in graphene nanoribbons

We study the interplay between the edge states and a single impurity in a zigzag graphene nanoribbon. We use tight-binding exact diagonalization techniques, as well as density functional theory calculations to obtain the eigenvalue spectrum, the eigenfunctions, as well as the dependence of the local density of states (LDOS) on energy and position. We study the strictly zero-energy eigenfunctions using symmetry considerations, as well as tight-binding techniques. Moreover, we note that roughly half of the unperturbed eigenstates in the spectrum of the finite-size ribbon hybridize with the impurity state, and the corresponding eigenvalues are shifted with respect to their unperturbed values. The maximum shift and hybridization occur for a state whose energy is inverse proportional to the impurity potential, and give rise to an impurity peak in the DOS spectrum. We find that the interference between the impurity and the edge gives rise to peculiar modifications of the LDOS of the nanoribbon, in particular to oscillations of the edge LDOS. These effects depend on the size of the system, and decay with the distance between the edge and the impurity.     

Perturbation theory for the decay rate of eigenfunctions in the generalizedN-body problem

Simple examples are known where eigenfunctions decay faster than the usual upper bounds would lead one to believe. We develop aspects of the perturbation theory of the decay rate of eigenfunctions as measured by radial exponential weights. We show that generically (in a Baire category sense) eigenfunction decay rates are governed by the lowest threshold.Supported in part by NSF grant DMS-8807816     

希尔伯特空间有限区域成像积分算子的谱

本文讨论了希尔伯特空间中有限区域成像积分算子的本征值和本征函数,研究了非对称核的情形,并用微扰和有限秩方法计算了衍射受限和离焦情况下的本征值和本征函数。     

Spontaneous emission from photonic crystals: full vectorial calculations

Quantum electrodynamics of atom spontaneous emission from a three-dimensional photonic crystal is studied in a full vectorial framework. The electromagnetic fields are quantized via solving the eigenproblem of photonic crystals with use of a plane-wave expansion method. It is found that the photon density of states and local density of states (LDOS) with a full band gap vary slowly near the edge of band gap, in significant contrast to the singular character predicted by the previous isotropic model. Therefore, the spontaneous emission can be solved by conventional Weisskopf-Wigner approximate theory, which yields a pure exponentially decaying behavior with a rate proportional to the LDOS.     

Interatomic Auger processes and the density of states

The correlation between the line shape of Auger peaks and the density of states near the surface has been the subject of recent controversy. In certain cases, it has been possible to obtain the density of states by numerical deconvolution of a KVV peak (Amelio, 1970) or directly using a KLV peak (Cardona et al., 1973). However, the extension of this technique to transition metals (Cu, Zn) has encountered serious difficulties, related to the perturbation created by the presence of localized charges either in the initial or in the final state, although it is not yet clear why this perturbation is strong only in certain cases. The purpose of the present communication is to show a series of results that can throw some light on the abovementioned problem. The main point is that Auger processes of interatomic type, as those occurring in the INS technique of Hagstrum, are free of these perturbations. Recently, the authors have studied the line shape of the Auger peaks of O, C, N and S adsorbed on Cu, Ni and Fe. These results show that only that part of the Auger structure originated by interatomic transitions between substrate and adsorbate atoms can be related to the local density of states (LDOS). The rest of the structure, due to normal intraatomic processes, is dominated by the spectral terms in the final configuration of the ion. This new interpretation allows a separation of perturbation effects and clarifies the contribution of the LDOS to the peak line shape. In this communication, we present the line shape analysis of the L_2,3 VV and KVV Auger peaks of Mg and O in MgO. Due to the strong ionic character of this compound, the L_2,3 VV peak of Mg⁺⁺ is mainly due to interatomic processes between Mg⁺⁺ and O⁼ ions, whereas the KVV peak of O is mainly due to interatomic processes. This analysis shows that good agreement exists between the L_2,3VV Mg⁺⁺ Auger peak and the self-convolution of MgO density of states, whereas the KVV Auger peak of O⁼ is dominated by the spectral terms of the final configuration. Only a small peak in the high energy side of the latter peak can be related to the density of states and could be interpreted as an interatomic transition between two neighboring oxygen ions, in agreement with the interpretation given by others.     

Direct comparison between potential landscape and local density of states in a disordered two-dimensional electron system

The local density of states (LDOS) of the adsorbate-induced two-dimensional electron system (2DES) on n-InAs(110) is studied by scanning tunneling spectroscopy. In contrast to a similar 3DES, the 2DES LDOS exhibits 20 times stronger corrugations and rather irregular structures. Both results are interpreted as consequences of weak localization. Fourier transforms of the LDOS reveal that the k values of the unperturbed 2DES still dominate the 2DES, but additional lower k values contribute. To clarify the origin of the LDOS patterns, we measure the potential landscape of the 2DES area. We use it to calculate the expected LDOS and find reasonable agreement between calculation and experiment.     

β-NaGdF4纳米球中Tb3+-Er3+下转换对局域态密度的尺寸效应

以Span 80 为模板,采用水热法合成了不同尺寸（4.7~115.5 nm）的β-NaGdF₄：1%Tb³⁺,1%Er³⁺ 纳米晶（NCs）。在Rayleigh限（粒子尺寸小于跃迁波长）下,研究了纳米晶尺寸对局域态密度（Local density of states,LDOS）的影响以及镶嵌在β-NaGdF₄纳米球中的Tb³⁺-Er³⁺的辐射和无辐射特性,进一步揭示下转换过程的物理机制。基于Tb³⁺-Er³⁺ 处在 β-NaGdF₄纳米球中的模型,用Green函数方法计算了Tb³⁺-Er³⁺ 发射体的自发发射速率。在介电纳米球内,Tb³⁺-Er³⁺发射体的LDOS没有显著的变化。在小尺寸（R<<λ）介电纳米球外,按照Chew的理论,发现LDOS有一个类-Gauss分布。如果R>35 nm（在本实验条件下）,介电纳米球外则只能观测到LDOS 的下降边,LDOS与局域场强的平方E²成正比,因而LDOS的类高斯分布出现的原因应归于小尺寸发射体与局域场相互作用的增强。通过计算纳米晶尺寸与体材料自发辐射速率的比值可直接确定纳米材料中的填充因子。     

Frequency shift of sources embedded in finite two-dimensional photonic clusters

The frequency (Lamb) shift and local density of states (LDOS) in two-dimensional photonic crystals composed of a cluster of infinitely long circular cylinders is calculated classically using the radiation reaction mechanism. We investigate the frequency shift and LDOS as a function of the size of the cluster and show that, at the edges of the band gap, both quantities can be large and increase in magnitude with cluster size. We explain this in terms of poles of a scattering operator and also show that both the Lamb shift and LDOS are sensitive functions of the shape of the cluster.     

Eigenfunctions and eigenvalues of the Wigner operator

It can be proved, that the Wigner operator, which results from the Quantum-Mechanical foundation of Bopp, accepts as eigenvalues the differences of the eigenvalues of two equivalent Schrödinger equations. The eigenfunctions result with the help of a Fourier transform in the phase space of the corresponding eigenfunctions of the Schrödinger equations.     

Spontaneous-emission control by local density of states of photonic crystal cavity

The local density of states (LDOS) of two-dimensional square lattice photonic crystal (PhC) defect cavity is studied.The results show that the LDOS in the centre is greatly reduced,while the LDOS at the point off the centre (for example,at the point (0.3a,0.4a),where a is the lattice constant) is extremely enhanced.Further,the disordered radii are introduced to imitate the real devices fabricated in our experiment,and then we study the LDOS of PhC cavity with configurations of different disordered radii.The results show that in the disordered cavity,the LDOS in the centre is still greatly reduced,while the LDOS at the point (0.3a,0.4a) is still extremely enhanced.It shows that the LDOS analysis is useful.When a laser is designed on the basis of the square lattice PhC rod cavity,in order to enhance the spontaneous emission,the active materials should not be inserted in the centre of the cavity,but located at positions off the centre.So LDOS method gives a guide to design the positions of the active materials (quantum dots) in the lasers.     

Frequency shift of sources embedded in finite two-dimensional photonic clusters

The frequency (Lamb) shift and local density of states (LDOS) in two-dimensional photonic crystals composed of a cluster of infinitely long circular cylinders is calculated classically using the radiation reaction mechanism. We investigate the frequency shift and LDOS as a function of the size of the cluster and show that, at the edges of the band gap, both quantities can be large and increase in magnitude with cluster size. We explain this in terms of poles of a scattering operator and also show that both the Lamb shift and LDOS are sensitive functions of the shape of the cluster.     

A new approach to the Schrödinger equation with rational potentials

A new analytic theory is established for the Schrödinger equation with a rational potential, including a complete classification of the regular eigenfunctions into three different types, an exact method of obtaining wavefunctions, an explicit formulation of the spectral equation (3 x 3 determinant) etc. All representations are exhibited in a unifying way via function-theoretic methods and therefore given in explicit form, in contrast to the prevailing discussion appealing to perturbation or variation methods or continued-fraction techniques. The irregular eigenfunctions at infinity can be obtained analogously and will be discussed separately as another solvable case for singular potentials.     

Analytical results for random walk persistence

In this paper, we present a detailed calculation of the persistence exponent straight theta for a nearly Markovian Gaussian process X(t), a problem initially introduced elsewhere in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. Resummed perturbative and nonperturbative expressions for straight theta are derived, which suggest a connection with the result of the alternative independent interval approximation. The perturbation theory is extended to the calculation of straight theta for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and nonperturbative expressions for the persistence exponent straight theta(X0), describing the probability that the process remains larger than X(0)sqrt[].     

Local density of states around two nonmagnetic impurities in cuprate superconductors

The local density of states (LDOS) around two nonmagnetic impurities which are located at different sites is studied within the two-dimensional t-J-U model. The order parameters are determined in a self-consistent way with the Gutzwiller projected mean-field approximation and the Bogoliubov-de Gennes theory. When the two impurities are located one or two sites away, we find the superconductivity coherence peaks are more strongly suppressed and the zero-energy peak (ZEP) has split into two peaks. Whereas when the two impurities are located next to each other, the ZEP vanished, and LDOS does not change a lot compared with the case away from the impurities.     

Tunnelling across a potential barrier

Generalized expressions for the tunnelling probability spectrum and the tunnelling time are derived for a potential barrier of arbitrary shape independent of the use of the perturbation theory. A method to obtain the tunnelling current density by solving the continuity equation in the barrier region is proposed.     

Eigenfunctions and Eigenvalues for a Scalar Riemann–Hilbert Problem Associated to Inverse Scattering

A complete set of eigenfunctions is introduced within the Riemann–Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schr?dinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation. Received: 17 December 1998/ Accepted: 21 July 1999     

On the application of Brillouin—Wigner perturbation theory to multireference configuration mixing

A posteriori corrections to state specific multireference configuration interaction are proposed which restore linear scaling with particle number. The corrections are based on an analysis of the Brillouin-Wigner perturbation theory, using a Lippmann-Schwinger-like equation.     

Calculation of the coupling constants of many-electron atoms with external fields using expansions in a discrete basis of sturmian-type virtual orbitals

Based on a generalization of the method of Sturmian expansions to the case of many-electron systems, a fundamentally new approach was developed for the construction of a complete set of intermediate states in many-body perturbation theory thereby avoiding integration over states of the continuous spectrum. A complete system of eigenfunctions of the generalized eigenvalue problem for the one-electron Hartree-Fock equation with a factor before the energy operator and fixed values of the Lagrange multipliers is used as a basis of single-particle virtual orbitals for an initial approximation to the eigenfunctions of the Hartree-Fock self-consistent field method. In the case of bound states, the spectrum of the effective operator is purely discrete, thus permitting one to completely eliminate integration over continuum virtual orbitals in applying methods of many-body perturbation theory to atomic and molecular calculations for a number of cases. As a consequence, one manages to avoid a number of the technical difficulties associated with inclusion of such states. The possibilities of the developed approach are illustrated in application to the investigation of the interaction of many-electron atoms with external fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 73–85, August, 1990.