| m | Partial wave treatment for two-dimensional Coulomb-scattering and Regge pole

The symmetry and |m| partial-wave analysis for two-dimensional (2D) Coulomb-scattering is investigated. As a function of energyE, the |m| partial-wave scattering amplitudef _|m|(θ) is analytically continuated to the, negativeE (complexk) plane, and it is found that the bound state energy eigenvalues (E<0) are just located at the poles off _|m|(θ) on the positive imaginaryk axis as is expected. In addition, as a function of |m|,f _|m|(θ) is analytically continuated to the complex |m| plane, the bound state energy eigenvalues are just located at the poles off _|m|(θ) on the positive real |m| axis.     

局域模(实的及虚的)的一种新表述

本文给出了一种新的表述局域模(实的及虚的)的方法。我们引入了变形豫解式的迹P(z).它仅在z平面的实轴上有奇异性,孤立极点对应于实局域模。将P(E+iη)向下解析延拓至第二黎曼面,则其复极点对应于虚的(或不稳定的)局域模。复极点的实部是不稳定局域模的能量,虚部是它寿命的倒数,自然地得到了寿命为正的限制。此能量与寿命符合于相应共振散射的能量与宽度。本文还讨论了实局域模与虚局域模之间的关系。最后,比较了P(z)的奇异性与态密度的变化,并进一步阐明了寿命为正的及负的解之间的差别。指出,一般说来,只有寿命为正的解当作用强度增加时会变成实的局域模,负寿命的解是不会的。     

一类非定域位势的Regge行为

对一类非定域位势证明了S矩阵元S(λ,k)作为角动量变数λ的函数在Reλ>0半纯,且当λ沿正实轴或负虚轴趋于∞时(S(λ,k)-1)exp(-iπλ)→0。在稍严一些的假设下证明了S(λ,k)在带域|Imk|<μ内对动量变数k半纯。讨论了极点的位置,这类位势在右半平面的Regge极点不一定在第一象限内。     

Analytic properties of the T matrix for two Yukawa potentials

A study is made of the behavior of the residues at the Regge poles of the partial-wave T matrix on the mass shell as a function of the energy E for nonrelativistic scattering on an exponential potential and two Yukawa potentials. It is shown that the residues do not have left-hand cuts in the E plane.     

Two-Body T-Matrices Without Angular-Momentum Decomposition: Energy and Momentum Dependences

The two-body T-matrix is calculated directly as function of two vector momenta for different Malfliet-Tjon-type potentials. At a few hundred MeV projectile energy the total amplitude is quite a smooth function showing only a strong peak in forward direction. In contrast, the corresponding partial-wave contributions, whose number increases with increasing energy, become more and more oscillatory with increasing energy. The angular and momentum dependence of the full amplitude is studied and displayed on as well as off the energy shell as function of positive and negative energies. The behaviour of the T-matrix in the vicinity of bound-state poles and resonance poles in the second energy sheet is studied. It is found that the angular dependence of T exhibits very characteristic properties in the vicinity of those poles, which are given by the Legendre function corresponding to the quantum number either of the bound state or the resonance (or virtual) state. This behaviour is illustrated along numerical examples. Received May 29, 1997; revised October 17, 1997; accepted for publication December 28, 1997     

The Analytic Eigenvalue Structure of the 1+1 Dirac Oscillator

We study the analytic structure for the eigenvalues of the one-dimensional Dirac oscillator,by analytically continning its frequency on the complex plane.A twofold Riemann surface is found,connecting the two states of a pair of particle and antiparticle.One can,at least in principle,accomplish the transition from a positive energy state to its antiparticle state by moving the frequency continuously on the complex plane,without changing the Hamiltonian after transition.This result provides a visual explanation for the absence of a negative energy state with the quantum number n=0.     

态|Ψ(3)>q中广义电场分量的偶数次N次方Y压缩

构造了由多模复共轭虚相干态、多模复共轭虚相干态的相反态和多模复共轭相干态的线性叠加所组成的一种新型的三态叠加多模叠加态光场|Ψ⁽³⁾>_q.利用多模压缩态理论,研究了态|Ψ⁽³⁾>_q中广义电场分量的偶数次等幂次N次方Y压缩特性.结果发现:态|Ψ⁽³⁾>_q是一种典型的三态叠加多模非经典光场;当压缩次数N等于2P且P为2m’+1(m’=0,1,2,…,…)时,如果各模的初始相位φ_j(j=1,2,3,…,…,q)、态间的初始相位差(θ₁-θ₂)、(θ₁-θ₃)和(θ₂-θ₃),以及各单模相干态光场的总的平均光子数等分别满足各自的取值条件,则在这种情况下态|Ψ⁽³⁾>_q的广义电场分量可呈现出偶数次的广义非线性等幂次2(2m’+1)次方Y压缩效应.     

Quantum interference of Rashba-type spin-split surface state electrons

We studied the quantum interference of electrons in the Bi (p(x), p(y)) orbital-derived j = 1/2 spin-split surface states at Bi/Ag(111)√3 × √3 surfaces of 10 monolayer thick Ag(111) films on Si(111) substrates. Surface electron standing waves were observed clearly at the energy (E) below the intersection of the two spin-split downward dispersing parabola bands (E(x)). The E dependence of the standing wave pattern reveals the dispersion as the average of the two spin-split surface bands due to the interference between |(k + Δ), ↑> and |-(k - Δ), ↑> [or (|(k - Δ), ↓>) and |-(k + Δ), ↓>] states. In contrast, it was impossible to deduce the dispersion from the standing wave pattern at E ≥ E(x) because the surface electron cannot find its backscattered state with the same spin polarization.     

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+¹²C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padéapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.     

Grover量子搜索算法的一般化多相位匹配

一个量子系统将不可避免地受到不可预知的微扰影响,据此断定文献中的Grover量子搜索算法的实验实现是在三维复子空间中完成的.同时证明在二维复子空间中,对任意给定的初始态|γ0＞=cosβ0| α＞+sinβ0eiζ|β＞(β0是较小的正实数,ζ是任意的一个实数),存在解集Fj={(θj,θj-1,…,θ1),(φj,φj-1,…,φ1)}(整数j≥2)使得目标态能以100％的最大成功概率找到,其中相位旋转角θj和φl是不为2k’π的实数(1≤l≤j,k’为任意整数).如果只要求目标态以较高的成功概率找到,那么当一个无序数据库中目标态和非目标态的总个数足够大时,对于相对较小的正整数j,解集Fj可表示为j∑l=1θl=j∑l=1φl的形式.     

Justification of c-number substitutions in bosonic Hamiltonians

The validity of substituting a c-number z for the k=0 mode operator a(0) is established rigorously in full generality, thereby verifying one aspect of Bogoliubov's 1947 theory. This substitution not only yields the correct value of thermodynamic quantities such as the pressure or ground state energy, but also the value of |z|(2) that maximizes the partition function equals the true amount of condensation in the presence of a gauge-symmetry-breaking term. This point had previously been elusive.     

Exact Solution of the Pseudoharmonic Oscillator in the Space of Constant Positive Curvature

We present analytically the exact solution of the radial Schrödinger equation with the pseudoharmonic oscillator potential in constant positive curvature representation. Exact bound state eigenfunctions and eigenvalues obtained using factorization method. Finally, energy eigenvalues obtained here compared with the results of the theoretical methods in the limit of flat space.     

Saddles and trajectories,and poles and surface waves in heavy-ion elastic scattering

The terms I_m^±(θ) of the Poisson summation formula are used to describe and classify the oscillatory structures of the heavy-ion elastic cross sections coming from a strong-absorption model. The method of stationary phase, which relates these terms to branches of a deflection function, is shown to be inadequate and is replaced by the saddle-point method in the plane of complex angular momenta. The positions of the saddle points are found to be fairly independent of the parametrisation and define certain “active” regions which contribute to the scattering amplitude. The complex saddle points may be thought of as corresponding to complex trajectories in the nuclear potential. If the strong-absorption parametrisation possesses poles many saddle-point contributions can be simply expressed as the residues of the poles nearest the real axis. One of these leading parametric poles has an energy dependence similar to that of the dominant Regge pole generated by a complex optical potential. Poles lying above the real axis give rise to terms which may be thought of as surface waves, i.e. they are damped as they progress around the nuclear surface. These terms possess classically equivalent paths in the sense that they correspond to deflections less than π. Terms coming from poles below the real axis have no classical equivalent, increase as they progress and can, then, only be thought of as being diffractive. Some useful formulae for the cross section are given and the limiting case of a sharp cut-off is discussed.     

Binding Energy of an Exciton Bound to Ionized Acceptor in Quantum Dots

Binding energiesfor an exciton (X) trapped in the two-dimensional quantum dot by a negative ion located on the z axis at a distance from the dot plane are calculated by using the method of few-body physics.This configuration is called a barrier (A^-,X) center.The dependence of the binding energy of the ground state of the barrier (A^-,X)center on the electron-to-hole mass ratio for a few values of the distance d between the fixed negative ion on the z axis and the dot plane is obtained.We find that when d → 0,the barrier (A^-,X) center has not any bound state.We also studied the stability and binding energy of the ground state of the barrier (A^-,X) center in a parabolic quantum dot as a function of the distance d between the fixed negative ion on the z axis and the dot plane.     

Binding energy of an exciton bound to ionized donors in quantum dots

Binding energies for an exciton (X) trapped in the two-dimensional quantum dot by a positive ion located on the z axis at a distance d from the dot plane are calculated by using the method of few-body physics. This configuration is called a barrier (D⁺,X) center. The dependence of the binding energy of the ground state of the barrier (D⁺,X) center on the dot radius for a few values of the distance d between the fixed positive ion on the z axis and the dot plane is obtained. We find that when d<0.2nm the barrier (D⁺,X) center does not form a bound state.     

Exact solution of the Klein‐Gordon equation for the PT‐symmetric generalized Woods‐Saxon potential by the Nikiforov‐Uvarov method

The exact solution of the one‐dimensional Klein‐Gordon equation of the ????‐symmetric generalized Woods‐Saxon potential is obtained. The exact energy eigenvalues and wavefunctions are derived analytically by using the Nikiforov and Uvarov method. In addition, the positive and negative exact bound states of the s‐states are also investigated for different types of complex generalized Woods‐Saxon potentials.     

The scattering of the Manning-Rosen potential with centrifugal term

In this Letter the approximately analytical scattering state solutions of the l-wave Schrödinger equation for the Manning-Rosen potential are carried out by a proper approximation to the centrifugal term. The normalized radial wave functions of l-wave scattering states are presented and the calculation formula of phase shifts is derived. It is well shown that the poles of the S-matrix in the complex energy plane correspond to bound states for real poles and scattering states for complex poles in the lower half of the energy plane. We consider and verify two special cases: the l=0 and the s-wave Hulthén potential.     

Parseval Theory of Complex Wavelet Transform for Wavelet Family Including Rotational Parameters

A rotational parameter R_θ has been introduced to complex wavelet transform (CWT). The rotational CWT (RCWT) corresponds to a matrix element 〈ψ|U₂ (θ;μ;k)|F〉in the context of quantum mechanics, where U₂(θ;μ;k) is a two-mode rotational displacing-squeezing operator in the 〈η| representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac's representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.     

Analytic Structure and Power-Series Expansion of the Jost Matrix

For the Jost-matrix that describes the multi-channel scattering, the momentum dependencies at all the branching points on the Riemann surface are factorized analytically. The remaining single-valued matrix functions of the energy are expanded in the power-series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain an analytic expression for the Jost-matrix (and therefore for the S-matrix) near an arbitrary point on the Riemann surface (within the domain of its analyticity) and thus to locate the resonant states as the S-matrix poles. This approach generalizes the standard effective-range expansion that now can be done not only near the threshold, but practically near an arbitrary point on the Riemann surface of the energy. Alternatively, The semi-analytic (power-series) expression of the Jost matrix can be used for extracting the resonance parameters from experimental data. In doing this, the expansion coefficients can be treated as fitting parameters to reproduce experimental data on the real axis (near a chosen center of expansion E ₀) and then the resulting semi-analytic matrix S(E) can be used at the nearby complex energies for locating the resonances. Similarly to the expansion procedure in the three-dimensional space, we obtain the expansion for the Jost function describing a quantum system in the space of two dimensions (motion on a plane), where the logarithmic branching point is present.