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.     

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.     

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.     

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...     

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.     

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.     

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.     

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.     

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.     

Renormalization method for the quantum system of interacting resonances

The procedure of constructing quasi-energy functions and a quasi-energy spectrum on the basis of renormalization of the hamiltonian is considered for a nonlinear quantum system in an external periodic field.     

Numerical method for the time evolution of the Dirac equation

We propose an iterative procedure for the numerical evolution of Dirac equation. It is a fast and stable version of the coordinate-space predictor-corrector method and can be extended to (3+1) dimensions for the relativistic mean-field calculation of heavy-ion collisions. A detailed example is worked out for the case of the Bottcher-Strayer pair production mechanism in (1+1) dimensions.     

An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: The use of redundant primitive coordinates

A modified effective scaling frequency factor (ESFF) method that takes advantage of the potential energy distribution (PED) coefficients calculated in the basis of redundant primitive internal coordinates is presented. This approach is simpler and more flexible than that based on the natural internal coordinates. The sets of optimal scaling factors for routine 9- and 11-parameter ESFF calculations based on B3LYP/6-311G∗∗ force fields are derived from Baker’s training set of 30 molecules (660 frequencies). The calculated root-mean-square (RMS) deviations for all frequencies are 11.42 and 11.44 cm⁻¹ for 9- and 11-parameter scaling, respectively. They are somewhat lower than in the case of ordinary ESFF calculations. The new sets of factors seem to be particularly well suited for scaling of frequencies in the middle region of the vibrational spectra (1000-2500 cm⁻¹) - the RMS values in this range are 8.37 for 9-, and 9.56 cm⁻¹ for 11-parameter scaling. These values are to be compared with 9.20 and 10.29 cm⁻¹, respectively, calculated within the natural coordinates based ESFF formalism.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

The Dirac Sea

We give an alternate definition of the free Dirac field featuring an explicit construction of the Dirac sea. The treatment employs a semi-infinite wedge product of Hilbert spaces. We also show that the construction is equivalent to the standard Fock space construction.     

A method of solving the Dirac equation

Electromagnetic fields are found for which the Dirac equation reduces to a system of independent second-order differential equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp.41–43, December, 1980.