Degenerate asymptotic perturbation theory

Asymptotic Rayleigh-Schrödinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case. For this purpose we study the spectral properties ofm×m—matrix functionsA(κ) of a complex variable κ which have an asymptotic expansion εA _k κ ^k as τ→0. We show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections ofA(κ) can be obtained from the set {A _κ} by analytic perturbation theory. Special attention is given to the case whereA(κ) is Borel-summable in some sector originating from κ=0 with opening angle >π. Here we prove that the asymptotic series describe individual eigenvalues and eigenprojections ofA(κ) which are shown to be holomorphic inS near κ=0 and Borel summable ifA _k ^* =A _k for allk. We then fit these results into the scheme of Rayleigh-Schrödinger perturbation theory and we give some examples of asymptotic estimates for Schrödinger operators.     

Rayleigh-Schrödinger perturbation theory at large order for radial relativistic Hamiltonians using hypervirial and Hellmann-Feynman theories

Relativistic hypervirial and Hellmann-Feynman theorems are used to construct Rayleigh-Schrödinger expansions for eigenvalues of perturbed radial Dirac equations to aribitrary order. The method is very simple and flexible, requiring no matrix elements. Only the unperturbed energy is required as input. Any difficulties due to the presence of unperturbed continuum states are bypassed. Particular attention is paid to hydrogenic atoms with confining scalar potentials of the form W(r) = λr^q, q = 0, 1, 2, …. Continued fraction representations of these expansions reveal their Stieltjes behavior for q ⩾ 1 and Padé summability for q = 1, 2.     

Interface Fluctuations for the D = 1 Stochastic Ginzburg–Landau Equation with Nonsymmetric Reaction Term

We consider a Ginzburg–Landau equation in the interval [?ε^?1, ε^?1], ε>0, with Neumann boundary conditions, perturbed by an additive white noise of strength $\sqrt {\varepsilon } $ and reaction term being the derivative of a function which has two equal–depth wells at ±1, but is not symmetric. When ε=0, the equation has equilibrium solutions that are increasing, and connect ?1 with +1. We call them instantons, and we study the evolution of the solutions of the perturbed equation in the limit ε→0⁺, when the initial datum is close to an instanton. We prove that, for times that may be of the order of ε^?1, the solution stays close to some instanton whose center, suitably normalized, converges to a Brownian motion plus a drift. This drift is known to be zero in the symmetric case, and, using a perturbative analysis, we show that if the nonsymmetric part of the reaction term is sufficiently small, it determines the sign of the drift.     

On the Born-Oppenheimer expansion for polyatomic molecules

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh ² of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.     

一维无序二元固体中电子局域性质的研究

从单电子紧束缚模型的哈密顿量出发，格点能量随机取ε_A和ε_B，只计及格点之间的近程跳跃积分，建立了一维无序二元固体模型. 利用负本征值理论及无限阶微扰理论，对系统电子的本征值和本征态进行了数值计算. 结果表明与一定能量本征值对应的电子波函数只分布在系统的一定范围内，显示了其局域性. 借助传输矩阵方法，计算出电子的局域长度，讨论了局域长度随本征能量和无序度的变化关系，并研究了计入不同范围跳跃积分下，局域长度的变化特征. 关键词： 无序 二元固体 电子态 局域长度     

本征应变为线性函数的椭圆夹杂

对于平面中的椭圆夹杂,当本征(或无应力)应变ε_ij^*为位置x,y的线性函数时,夹杂内的拘束应变ε_ij^C亦为位置的线性函数。本文给出了全部的一级拘束系数。在椭圆的轴比β足够小时,我们还得到了夹杂外部场的解析表达式。本文结果可直接应用于处理一些实际问题,例如裂纹在弯曲加载条件下的行为,以及片状夹杂与微裂纹的相互作用等。 关键词：     

On the Eigenvalues of the Fermionic Angular Eigenfunctions in the Kerr Metric

In view of a result recently published in the context of the deformation theory of linear Hamiltonian systems, we reconsider the eigenvalue problem associated with the angular equation arising after the separation of the Dirac equation in the Kerr metric, and we show how a quasilinear first order PDE for the angular eigenvalues can be derived efficiently. We also prove that it is not possible to obtain an ordinary differential equation for the eigenvalues when the role of the independent variable is played by the particle energy or the black hole mass. Finally, we construct new perturbative expansions for the eigenvalues in the Kerr case and obtain an asymptotic formula for the eigenvalues in the case of a Kerr naked singularity.     

On quantum waveguide with shrinking potential

The spectrum of a Schrödinger operator in a multidimensional cylinder perturbed by a shrinking potential is considered. The phenomenon of new eigenvalues emerging from the threshold of the essential spectrum is studied and sufficient conditions for this phenomenon are given. Asymptotic expansions for these eigenvalues are found (if such eigenvalues exist).     

Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

We consider a quantum waveguide modelled by an infinite straight tube with arbitrary cross-section in n-dimensional space. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators “localized” in a certain sense. We study the asymptotic behaviour of the discrete spectrum of such system as the distance between the “supports” of localized perturbations tends to infinity. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We provide a list of the operators, which can be chosen as distant perturbations. In particular, the distant perturbations may be a potential, a second order differential operator, a magnetic Schrödinger operator, an arbitrary geometric deformation of the straight waveguide, a delta interaction, and an integral operator.     

Eigenwerte des Lippmann-Schwinger-Kerns für Yukawa- und Nukleon-Nukleon-Potentiale

In several papers we recently obtained simple high-energy asymptotic expansions for the solutions and eigenvalues of wave equations containing generalized superpositions of Yukawa potentials. In the present article we extend these investigations to the calculation of phase shifts and eigenvalues of the Lippmann-Schwinger kernel. We also calculate corresponding Padé approximants and illustrate, by means examples, their usefulness, even in regions of low energies.     

低温氮化硅薄膜的介电性能研究

研究了微波电子回旋共振等离子体化学汽相沉积低温氮化硅薄膜在5—10⁶Hz频率范围内的介电性能．由于低温氮化硅薄膜为具有分形结构的纳米非晶薄膜，致使氮化硅薄膜的介电谱、损耗谱在低频区和高频区具有两种不同的分布规律．在低频区介电谱具有ε′∝ω^n－1₁的关系，ｎ₁在0.82—0.88之间，是电子跳跃导电的结果；在高频区介电谱具有ε′∝ω^n-1₂的关系，n₂在0 关键词：     

Multidimensional perturbation theory for nonseparable wave equations

We show that powerful procedures can be developed for deriving explicit asymptotic expansions for the solutions and eigenvalues of a large class of nonseparable wave equations. In particular, these expansions allow an investigation of the large-order behaviour.     

Convergence of the multipole results for first order orientation dependent intermolecular forces using NH3-H+ as a model interaction

The NH₃-H⁺ interaction is used as a model for discussing the usefulness of energy multipole expansions for interactions where the first order energy plays an important role. The multipole results for the first order energy are analysed formally and are compared numerically with the non-expanded first order energy for a variety of NH₃-H⁺ relative configurations. The results are used to discuss the limitations of the multipole expansion of the first order energy which can be very severe for some NH₃-H⁺ collision trajectories.     

电子极化对氟化钙离子晶体的弹性系数、压电系数和介电常数的影响

本文用黄昆的方法计算了电子极化对氟化钙离子晶体的弹性系数c₁₂-c₄₄的偏离的贡献,以及对静电与光学介电常数差的影响,其结果在数量级和方向上与实验结果符合。     

1/N series for quantum anharmonic oscillator eigenvalues and green functions

The large orders of the 1/N series for the O(N)-symmetric anharmonic oscillator are computed here in the functional approach. We explicitly give the Kth order of the 1/N series up to O(K^?1) for all energy eigenvalues and for the two-point Green function. Our results are derived considering solutions in a finite (but large) time interval (euclidean time).     

A New Version of the Fast Multipole Method for Screened Coulomb Interactions in Three Dimensions

We present a new version of the fast multipole method (FMM) for screened Coulomb interactions in three dimensions. Existing schemes can compute such interactions in O(N) time, where N denotes the number of particles. The constant implicit in the O(N) notation, however, is dominated by the expense of translating far-field spherical harmonic expansions to local ones. For each box in the FMM data structure, this requires 189p⁴ operations per box, where p is the order of the expansions used. The new formulation relies on an expansion in evanescent plane waves, with which the amount of work can be reduced to 40p²+6p³ operations per box.     

Double wells: Perturbation series summable to the eigenvalues and directly computable approximations

We give a rigorous proof of the analyticity of the eigenvalues of the double-well Schrödinger operators and of the associated resonances. We specialize the Rayleigh-Schrödinger perturbation theory to such problems, obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method. By an approximation we obtain new series expansions directly computable, still summable, which, in the case of the Herbst-Simon model, can be given in an explicit form.Partially supported by Ministero della Pubblica Istruzione     

原子核的单粒位阱(Ⅰ)——一个定理

本文证明了一个定理并讨论了它的一些应用。定理给出了单粒子能量与分离能的严格关系。     

Application of the quantum mechanical hypervirial theorems to even-power series potentials

The class of the even-power series potentials,V(r)=-D+∑ _k-0 ^∞ V_kλ^kr^2k+2,V ₀=ω²>0is studied with the aim of obtaining approximate analytic expressions for the nonrelativistic energy eigenvalues, the expectation values for the potential and kinetic energy operators, and the mean square radii of the orbits of a particle in its ground and excited states. We use the hypervirial theorems (HVT) in conjunction with the Hellmann-Feynman theorem (HFT), which provide a very powerful scheme for the treatment of the above and other types of potentials, as previous studies have shown. The formalism is reviewed and the expressions of the above-mentioned quantities are subsequently given in a convenient way in terms of the potential parameters, the mass of the particle, and the corresponding quantum numbers, and are then applied to the case of the Gaussian potential and to the potentialV(r)=−D/cosh²(r/R). These expressions are given in the form of series expansions, the first terms of which yield, in quite a number of cases, values of very satisfactory accuracy.     

Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ ^∞-norm of the corresponding eigenvectors is of order O(N ^−1/2), modulo logarithmic corrections. The upper bound O(N ^−1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements. Supported by Sofja-Kovalevskaya Award of the Humboldt Foundation. On leave from Cambridge University, UK. Partially supported by NSF grant DMS-0602038.