Determination of bound states and resonances of the Debye-potential by complex scaling

For a statically screened Coulomb-potential, two particle bound and resonant states are determined which correspond to complex poles of the S-matrix. The complex scaling method is used in order to calculate both the real eigenvalues of the Hamiltonian and the complex resonances in the same way. The eigenfunctions are described by complex power series; for the coefficients a recursion formula is given.     

Above barrier Dirac multiple scattering and resonances

We extend an above barrier analysis made with the Schrödinger equation to the Dirac equation. We demonstrate the perfect agreement between the barrier results and back to back steps. This implies the existence of multiple (indeed infinite) reflected and transmitted wave packets. These packets may be well separated in space or partially overlap. In the latter case interference effects can occur. For the extreme case of total overlap we encounter resonances. The conditions under which resonance phenomena can be observed is discussed and illustrated by numerical calculations.     

Conditions for ponderomotive resonances in the Kapitza–Dirac effect

By applying a nonperturbative quantum electrodynamic theory, we study ponderomotive resonances when an electron beam is scattered by a standing photon wave. Our study shows that the ponderomotive parameter u_p, the ponderomotive energy per laser-photon energy, for each of the two traveling laser modes possesses a minimum value ℏω/(mec2). Ponderomotive resonances occur only when the ratio of the laser photon energy to the electron rest-mass energy is a fraction, where the denominator is twice the square of a positive integer and the numerator is the total ponderomotive number, which is also a positive integer.     

Classical scaling theory of quantum resonances

The quantum resonances occurring with delta-kicked particles are studied with the help of a fictitious classical limit, establishing a direct correspondence between the nearly resonant quantum motion and the classical resonances of a related system. A scaling law which characterizes the structure of the resonant peaks is derived and numerically demonstrated.     

谐振子基在复标度方法中的应用

在利用薛定谔方程求解共振态能量的过程中,成功的将谐振子基应用于复标度方法,求解出共振态的能量公式,并以一个比较成熟的势作为检验势,得出比较精确的结果,也作出共振态能量在复能量坐标系中的能量分布.对其中的两个参数基数N和转动角θ进行讨论与分析,验证了共振态的一个原理:在对共振态的计算过程中,计算参数的改变不会影响共振态的位置.     

SS-HORSE method for studying resonances

A new method for analyzing resonance states based on the Harmonic-Oscillator Representation of Scattering Equations (HORSE) formalism and analytic properties of partial-wave scattering amplitudes is proposed. The method is tested by applying it to the model problem of neutral-particle scattering and can be used to study resonance states on the basis of microscopic calculations performed within various versions of the shell model.     

谐振子基在复标度方法中的应用

在利用薛定谔方程求解共振态能量的过程中，成功的将谐振子基应用于复标度方法，求 解出共振态的能量公式，并以一个比较成熟的势作为检验势，得出比较精确的结果，也作出共振态能量在复能量坐标系中的能量分布。对其中的两个参数基数N和转动角µ进行讨论与分析，验证了共振态的一个原理：在对共振态的计算过程中，计算参数的改变不会影响共振态的位置。     

Integration method for the Dirac equation

An integration method for the Dirac equation is proposed. The method, based on diagonalization, reduces the problem to one of integration of independent second-order differential equations.     

Relativistic extension of the complex scaled Green's function method for resonances in deformed nuclei

The European Physical Journal A - It is shown that the excellent scaling of the elliptic flow found for all centralities, species and energies from RHIC to the LHC for pT less than the saturation...     

复标度方法对原子核单粒子共振态的研究

在相对论平均场理论框架下, 利用复标度方法研究了 Zr 同位素的单粒子共振问题. 以 ¹²²Zr 为例, 演示了复标度方法确定共振态的具体步骤, 确定了 ¹²²Zr 所有可能共振态的能量和宽度, 以及相应共振态的复标度波函数, 并与耦合常数的解析延拓方法进行了比较.在此基础上, 进一步系统研究了 Zr 同位素的共振问题, 获得了与散射相移方法一致的结果.     

单粒子共振态的实稳定方法研究

基于实稳定方法的基本思想，在坐标空间求解了一维势阱和三维球对称势阱中的单粒子共振态，得到的共振能量、宽度和波函数等与散射相移法以及耦合常数的解析延拓方法的结果一致. 关键词： 共振态 实稳定方法 相移 共振宽度     

The determinantal method for bound states and resonances of three-particle systems

Bound state energies, positions and widths of resonances of two-particle systems may be calculated as zeros of an analytic function of the energy, the modified Fredholm determinant of the Lippmann-Schwinger equation. This generates degenerate perturbation theory particularly easily. The method is generalized to the threebody problem. Here too, the bound states and resonances appear as zeros of an analytic function of the energy, the modified Fredholm determinant of a square-integrable kernel. It is proved that the multiplicity of a zero equals the degeneracy of the corresponding eigenvalue.Invited talk at the symposium Theory of lightest nuclei, Liblice, Czechoslovakia, May 1974.Work supported in part by the National Science Foundation.     

On the applicability of the exterior complex scaling method for scattering problems including Coulombic potentials

We study some formal aspects of the exterior complex scaling (ECS) approach when implemented for both short and long-range potentials. In particular, we focus on the inconsistencies related to the requirement of an artificial cut-off of the potential in order to avoid exponential divergencies due to the complex rotation. For the pure two-body Coulomb potential we demonstrate analytically and numerically that the ECS inner solution is indeed the correct one, thus reinforcing the method; the extraction of the transition amplitude, however, remains problematic. We also show that a consistent application of the ECS method requires a distorted wave formulation, and two variants are proposed. Finally, we will propose an approach equivalent to the original ECS but that avoids all formal difficulties. It is based on performing the complex rotation on the basis functions rather than on the driven equation itself, and makes use of Sturmian functions with appropriately chosen outgoing boundary conditions. Our proposal differs from one of the original versions of the ECS method, through the use of physically based basis functions rather than pure numeric ones.     

Finite-size scaling in complex networks

A finite-size-scaling (FSS) theory is proposed for various models in complex networks. In particular, we focus on the FSS exponent, which plays a crucial role in analyzing numerical data for finite-size systems. Based on the droplet-excitation (hyperscaling) argument, we conjecture the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process, all of which are confirmed reasonably well in numerical simulations.     

Dirac equation in low dimensions: The factorization method

We present a general approach to solve the (1+1)$(1+1)$ and (2+1)$(2+1)$-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials.     

The WKB method for the Dirac equation with the vector and scalar potentials

The WKB approximation is developed for the Dirac equation with the spherically symmetrical vector and scalar potentials. The relativistic wavefunctions are constructed, new quantization rule containing the spin-orbital interaction is obtained. For spherically symmetrical model of the Stark effect the quasi-classical spectrum of relativistic hydrogen-like atom is calculated. Application of the WKB method to the mass spectrum of the hydrogen-like quark systems was done.