Perturbation theory for velocity-dependent potentials

In the presence of a velocity-dependent Kisslinger potential, the partial-wave, time-independent Schr?dinger equation with real boundary conditions is written as an equation for the probability density. The changes in the bound-state energy eigenvalues due to the addition of small perturbations in the local as well as the Kisslinger potentials are determined up to second order in the perturbation. These changes are determined purely in terms of the unperturbed probability density, the perturbing local potential, as well as the Kisslinger perturbing potential and its gradient. The dependence on the gradient of the Kisslinger potential stresses the importance of a diffuse edge in nuclei. Two explicit examples are presented to examine the validity of the perturbation formulas. The first assumes each of the local and velocity-dependent parts of the potential to be a finite square well. In the second example, the velocity-dependent potential takes the form of a harmonic oscillator. In both cases the energy eigenvalues are determined exactly and then by using perturbation theory. The agreement between the exact energy eigenvalues and those obtained by perturbation theory is very satisfactory. Received: 24 May 2002 / Accepted: 15 July 2002 / Published online: 3 December 2002 RID="a" ID="a"e-mail: mij@hu.edu.jo Communicated by V. Vento     

Phase-function method for velocity-dependent potentials

We have adapted the phase-function method for studying on- and off-shell properties of velocity-dependent potentials. The main result presented in this paper is an ansatz for the interpolatingT-matrix function (on- or off- the energy-shell as the case may). Based on this ansatz we have presented an efficient method for computing the off-shell extension function which plays a role in the theories of three particle system. We have demonstrated this by means of a model calculation.     

Path integration for velocity-dependent potentials

We extent the prodistribution definition of path integrals to include Lagrangians with velocity-dependent potentials. We use Cameron-Martin transformations to evaluate a large part of a path integral exactly and give techniques for evaluating the remaining terms of the semi-classical expansion of the path integral. The Fredholm determinants, associated with these transformations, are evaluated explicitly in terms of Jacobi matrices defined by the classical system.     

Linearly velocity-dependent potentials and the Lorentz-Lorenz correction

The Lorentz-Lorenz correction for linearly velocity-dependent potentials saturates at the second-order term. This sheds some new light onto the role of tensor potentials in the relativistic Dirac approach to proton nucleus scattering.     

Generalized semiclassical optical model and velocity-dependent potentials

A procedure is developed for the construction of a complex potential in a generalized semiclassical optical model for molecular collisions involving internal nuclear degrees of freedom. The procedure involves a local approximation on the exact quantum optical potential, which is an integral operator over an energy-dependent, non-local kernel. The resulting potential for the model is velocity-dependent. The classical limit of transition amplitudes is obtained from complex-valued classical trajectories whose equations of motion are derived with this potential. The potential contains coordinate integrals over the kernel, and various approximations for the kernel are discussed. Sample calculations are carried out for collinear atom-diatom collisions.     

Nonlocal gravitational models and exact solutions

A nonlocal gravity model with a function f (□^?1 R), where □ is the d’Alembert operator, is considered. The algorithm, allowing to reconstruct f(□^?1 R), corresponding to the given Hubble parameter and the state parameter of the matter, is proposed. Using this algorithm, we find the functions f(□^?1 R), corresponding to de Sitter solutions.     

On the relation between hard-core and velocity-dependent potentials

A canonical transformation relating hard-core and velocity-dependent nucleon-nucleon potentials is applied to the Srivastava potential and an equivalent hard-core potential is found. It is shown that the deuteron photonuclear electric-dipole integrated and bremsstrahlung-weighted cross sections resulting from the two equivalent potentials are essentially the same. The reasons for this agreement suggest that differences between the two sets of cross sections may remain small in other nuclei employing this type of potential.     

Perturbation theory for isotropic velocity-dependent potentials: Bound-states case

Starting from the time-independent Schr?dinger equation we develop formulae for the changes in the bound-state energies in the presence of an isotropic, velocity-dependent perturbing potential. The corresponding changes in the wave functions are also obtained. Unlike the case of the standard perturbation theory, determination of the changes in the energy and the wave function of a state only requires knowledge of the unperturbed ground-state wave function in addition to the perturbing potential. Evaluations of the energy changes and the corresponding wave functions are given for two examples in the s-wave case.     

Comparative analysis of approximate and exact models in inflationary cosmology

A method of constructing and analyzing exact solutions for inflationary cosmology models with a self-action scalar by introducing an effective self-action potential is suggested. On the basis of exact solutions for complete and “shortened” equations obtained in the “slow-descent” approximation, their comparative analysis is made. It is shown that the results obtained for approximate models that are conventionally used for comparison with experimental data may differ greatly from those for exact models because of the structural instability of models with inflation. Ul'anovsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 14–20, January, 2000.     

Lifshitz theory-— exact or approximate statistical mechanics?

We show that the Lifshitz and related theories of intermolecular interactions are equivalent to perturbation theory, and therefore subject to the usual restrictions of the latter.     

Effect of the velocity-dependent potentials on the energy eigenvalues of the Morse potential

We investigate the effect of the isotropic velocity-dependent potentials on the bound state energy eigenvalues of the Morse potential for any quantum states. When the velocity-dependent term is used as a constant parameter, ρ(r) = ρ ₀, the energy eigenvalues can be obtained analytically by using the Pekeris approximation. When the velocity-dependent term is considered as an harmonic oscillator type, ρ(r) = ρ ₀ r ², we show how to obtain the energy eigenvalues of the Morse potential without any approximation for any n and ℓ quantum states by using numerical calculations. The calculations have been performed for different energy eigenvalues and different numerical values of ρ ₀, in order to show the contribution of the velocity-dependent potential on the energy eigenvalues of the Morse potential.     

WKB and exact wave functions for inverse power-law potentials

We compare WKB and exact wave functions for inverse power-law potentials and derive a universal algebraic formula reproducing reflection phases and scattering phase shifts with uniform high accuracy for any wave length. Appropriately modified quantization gives competitive accuracy for the spiked harmonic oscillator. The asymptotic accuracy of WKB wave functions is discussed.     

非局域表面暗孤子及其稳定性分析

非局域体介质中的暗孤子及表面亮孤子由于在光通信领域的潜在应用而受到极大关注,然而到目前为止却没有对非局域表面暗孤子的研究.在线性介质和非局域非线性介质的分界面上,数值模拟得到了1+1维非局域基态和二阶表面暗孤子,研究了它们的波形与传播常数和介质非局域程度的关系,基于它们的稳定性分析进行了理论推导和数值模拟.稳定性分析结果表明:1+1维非局域基态表面暗孤子在其存在区域总是稳定的,而二阶表面暗孤子是区域不稳定的,其不稳定区域的宽度与传播常数以及介质的非局域程度有关系,且受传播常数的影响更大.加噪声的初始输入传输图验证了稳定性分析结果的正确性.     

Rosen-Morse potentials for relativistic spinless particles; approximate solutions

We deal with the D-dimensional radial Klein-Gordon equation for Rosen-Morse type potential with unequal scalar and vector potentials. We apply a proper approximation to the centrifugal term and, by proposing an ansatz solution to the resulting equation, we obtain the bound-state solution. To check the accuracy of our results, we compare our obtained quasi-analytical energies with the exact numerical ones.     

Dirac equation exact solutions for generalized asymmetrical Hartmann potentials

In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of vector and scalar generalized Hartmann potentials. This is done provided the vector potential is equal to or minus the scalar potential. The cases of some quasi-exactly solvable and Morse-like potentials are briefly commented.     

Non-hermitian exact local bosonic algorithm for dynamical quarks

We present an exact version of the local bosonic algorithm for the simulation of dynamical quarks in lattice QCD. This version is based on a non-hermitian polynomial approximation of the inverse of the quark matrix. A Metropolis test corrects the systematic errors. Two variants of this test are presented. For both of them, a formal proof is given that this Monte Carlo algorithm converges to the right distribution. Simulation data are presented for different lattice parameters. The dynamics of the algorithm and its scaling in lattice volume and quark mass are investigated.