Nonlocal Separable Potential Research Articles

Low and intermediate energy cross section analysis of p-p and p-d elastic scattering using modified phase amplitude method for electromagnetic distorted nuclear interaction

The traditional phase function method is regarded as a resourceful tool for computing scattering phase shifts for quantum mechanical potential scattering involving local interactions. For the local plus a nonlocal potential the conventional phase function method does not hold good. In such a situation one has to modify it. The modified variable phase method is adapted here to treat the Manning-Rosen (MR) plus Graz nonlocal separable potential for constructing an exact expression for the phase function with rigorous inclusion of the electromagnetic effect. The merit of our analytical result is demonstrated by computing elastic scattering phase parameters and differential scattering cross sections for proton-proton (p − p) and proton-deuteron (p − d) systems which are in sensible agreement with previous works.

$$\boldsymbol{p}\boldsymbol{-}^{\mathbf{12}}$$C and $$\boldsymbol{p}\boldsymbol{-}\boldsymbol{d}$$ Scattering within the Separable Model of Interaction

Differential equation approach is adapted to construct exact analytical expressions for the regular, Jost, and physical states for Hulthen plus separable nonlocal potential. The Jost function and the Fredholm determinant for physical boundary condition are exploited to compute scattering phase shifts for proton–deuteron and proton–carbon systems within this two-body model of interaction. Excellent agreement in phase shift values with other sophisticated calculations is obtained.

The Fredholm determinant for Hulthén-distorted non-local separable potential: Application to $$ \alpha {-}\alpha $$ elastic scattering

Exact analytical expression of the Fredholm determinant with outgoing wave boundary condition for motion in Hulthen-distorted non-local separable potential is constructed and expressed in the maximum reduced form. Using boundary conditions (regular and irregular), two approximate energy-dependent interactions corresponding to the parent non-local potential are also constructed. The phase shifts for the $$ \alpha $$ – $$\alpha $$ elastic scattering are computed by using (i) exact expression for the Fredholm determinant and (ii) energy-dependent local interactions by exploiting the phase function method. The merits of our constructed equivalent energy-dependent potentials are judged by comparing the $$ \alpha $$ – $$\alpha $$ elastic scattering phases with our exact calculation and standard data.

Generating Energy-Dependent Potentials

Energy-momentum-dependent potentials corresponding to a separable nonlocal potential and a separable nonlocal potential plus a local potential are constructed to study nucleon-nucleon and nucleus-nucleus systems. The constructed local potentials developed via two different approaches either have no hard core or a quasi-hard core. Elastic scattering phase shifts are computed for the proton-proton and the alpha-alpha systems by using the phase function method and are found to be in good agreement with experimental data.

On the Alpha-Carbon-12 Elastic Scattering

A closed form expression for the Fredholm determinant is constructed for motion in Coulomb-distorted nonlocal separable potential and expressed it in its maximal reduced form. We demonstrate the usefulness of our constructed expression by means of a model calculation.

Potentials and phase shifts for nucleon–light nuclei systems

Two simple models based on the Coulomb-distorted phase function and supersymmetry-inspired factorisation methods are adapted to deal with the nucleon–light nuclei elastic scattering at low energies. The first one is associated with the derivation of a closed-form expression of the scattering phase shift for motion in Coulomb-distorted separable non-local potentials. The second one deals with the development of an energy-dependent phase equivalent local potential to the non-local one for s-wave and its subsequent generation of higher partial wave interactions through the formalism of supersymmetric quantum mechanics. The usefulness of our models is demonstrated through the computation of $$\upalpha $$ –nucleon scattering phase shifts at low energies up to partial waves $$\ell = 2$$ . Certain energy-dependent correction factors are also incorporated into energy-dependent higher partial wave potentials to achieve an excellent agreement with the standard data.

Ground state atoms confined in a real Rydberg and complex Rydberg-Scarf II potential

In this work, a system of two ground state atoms confined in a one-dimensional real Rydberg potential was modeled. The atom-atom interaction was considered as a nonlocal separable potential (NLSP) of rank one. This potential was assumed because it leads to an analytical solution of the Lippmann-Schwinger equation. The NLSPs are useful in the few body problems that the many-body potential at each point is replaced by a projective two-body nonlocal potential operator. Analytical expressions for the confined particle resolvent were calculated as a key function in this study. The contributions of the bound and virtual states in the complex energy plane were obtained via the derived transition matrix. Since the low energy quantum scattering problems scattering length is an important quantity, the behavior of this parameter was described versus the reduced energy considering various values of potential parameters. In a one-dimensional model, the total cross section in units of the area is not a meaningful property; however, the reflectance coefficient has a similar role. Therefore the reflectance probability and its behavior were investigated. Then a new confined potential via combining the complex absorbing Scarf II potential with the real Rydberg potential, called the Rydberg-Scarf II potential, was introduced to construct a non-Hermitian Hamiltonian. In order to investigate the effect of the complex potential, the scattering length and reflectance coefficient were calculated. It was concluded that in addition to the competition between the repulsive and attractive parts of both potentials, the imaginary part of the complex potential has an important effect on the properties of the system. The complex potential also reduces the reflectance probability via increasing the absorption probability. For all numerical computations, the parameters of a system including argon gas confined in graphite were considered.

Quantum scattering approach for investigation of two interacting atoms trapped in a one-dimensional Morse potential via Lippmann-Schwinger equation

In this work, a system including two neutral atoms confined to an external one-dimensional Morse potential was modelled. The problem can be relevant to cold atom physics, where neutral atoms may be effectively confined in radially tight tubes formed by optical lattices. The atom-atom interaction was considered as a nonlocal separable potential. Analytical expressions for wave-function as well as transition matrix were derived. The contributions of bound states and resonances in the complex energy plane were calculated. For numerical computations, the bound states in a system of argon gas confined in graphite were considered. Since the most important quantity in the low energy quantum scattering problems is “scattering length,” considering various values of Morse parameters, the behavior of this parameter was described versus the reduced energy.

Theoretical investigation of scattering of an atomic projectile confined in a harmonic surface potential

In this paper, scattering of a projectile atomic gas confined in an atomic harmonic surface is studied using the Lippmann–Schwinger. The nonlocal separable potential of rank one has been assumed between the projectile gas and surface, because this potential is useful to investigate the few-body systems. The analytical solution of Lippmann–Schwinger equation has been derived, and the scattering properties including transition and scattering matrices, phase shift, scattering amplitude and time delay are calculated analytically as a function of incident atomic gas energies.

Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties

In this work, the Faddeev equations for three-body scattering at arbitrary angular momentum are exactly solved and the transition matrices for some transition processes, including scattering and rearrangement channels are formulated in terms of free-particle resolvent matrix. A generalized Yamaguchi rank-two nonlocal separable potential has been used to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The partial-wave analysis for some transition processes in a three-particle system is suggested. The partial-wave three-particle transition matrix elements have been constructed via knowledge of the matrix elements of the free motion resolvent. The calculation of a number of scattering properties of interest of the system such as transition matrix and its poles (bound states and resonances) and consequently other related quantities like scattering amplitudes, scattering length, phase shifts and cross sections are feasible in a straightforward manner. Moreover, we obtain a new analytical expression for the third virial coefficient in terms of three-body transition matrix.

Binding energy of hydrogenic impurities in quantum dots under intense laser radiation

We calculate the binding energy of on- and off-center hydrogenic impurities in a parabolic quantum dot subjected to an intense high-frequency laser field. An exactly solvable model that replaces the actual Coulomb interaction with the donor by a non-local separable potential is introduced for calculating the binding energy. The separable potential allows us to solve the problem exactly and all calculations are carried out analytically. The action of the laser irradiation results in dressed Coulomb and confinement potentials. At low laser intensity the binding energy is found to decrease when the impurity is shifted away from the origin. At high laser intensity and strong confinement the opposite behavior is observed. We propose a simple one-dimensional model that explains the observed crossover.

Non-local separable solutions of two interacting particles in a harmonic trap

We calculate the energy levels of two particles trapped in a harmonic potential. The actual two-body potential, assumed to be spherically symmetric, is replaced by a projective operator (non-local separable potential) to determine the energy levels in a closed form. This approach overcomes the limitations of the regularized Fermi pseudopotential when the characteristic length of the two-body interaction potential is of the order of the size of the harmonic trap. In addition, we recover the results obtained with the Fermi pseudopotential when the length of the interaction is much smaller than the size of the trap.

Riemann surface approach to bound and resonant states for a nonlocal potential

The Riemann surface approach to bound and resonant states is extended to the case of a separable nonlocal potential that is constant on a certain domain of the inner region and vanishes in the rest of the domain. The approach consists in the construction of the Riemann surface \documentclass[12pt]{minimal}\begin{document}$R_g$\end{document}Rg of the S-matrix pole function k = k(g) over the g-plane, where g is the strength of the complex nonlocal potential. On the Riemann surface \documentclass[12pt]{minimal}\begin{document}$R_g$\end{document}Rg the pole function k = k(g) is single-valued and analytic. The branch points of the pole function k = k(g) and their k-plane images are determined and analyzed as a function of the position of the region of nonlocality. The Riemann surface of the S-matrix pole function is constructed. According to the Riemann surface approach to each bound or resonant state a sheet of the Riemann surface \documentclass[12pt]{minimal}\begin{document}$R_g$\end{document}Rg is associated. All the natural modes (bound and resonant states) of the system are identified and treated in a unified way. The nonlocal potential generates narrow resonant states that cannot be produced by a local potential.

Zero-range interaction in arbitrary dimensions and in the presence of external forces

We analyze zero-range interactions in arbitrary dimensions and in the presence of external forces in terms of non-local separable potentials. We show that in one-, two- and three-dimensional cases this approach is equivalent to the standard method with a renormalization operator. Our analysis indicates that the method of a projection operator is also useful for studying processes occurring under the influence of additional forces. As an example we present exact solutions for a particle interacting with a static electric field.

Partial-wave Scattering and Statistical Mechanics via the l-wave Non-local Separable Potential of Rank-two

The equilibrium statistical mechanics relations are shown to be related to T-matrix, which describes the scattering processes taking place in the thermodynamic system consisting of free particles and independent correlated pairs, interacting via the separable non-local potential of rank two in the lth partial wave. Thermodynamic properties are related to the correlated states, when making a pole expansion of the analytically continued momentum matrix element of R l (z), the difference between the resolvents of the interacting and free Hamiltonians. It is shown that local potentials equivalent to the nonlocal ones have an attractive part which is responsible for a bound state and negative values of some thermodynamic properties.

Modeling of Coulomb interaction in parabolic quantum wires

We study the exciton states in a parabolic quantum wire. An exactly solvable model is introduced for calculating the exciton state and the binding energy as a function of the radius of the quantum wire within the envelope-function approximation. In the calculation, we replace the actual Coulomb interaction between the electron and the hole by a Gaussian nonlocal separable potential and obtain closed expressions for both the envelope-function and the binding energy. Results are compared with those obtained by perturbative methods.

Scattering via a separable potential with higher angular momenta: application to statisticalmechanics

This paper presents a new formulation of the equilibrium and non-equilibrium statisticalmechanical properties of a gas consisting of free particles and independent correlated pairswith any angular momentum , interacting via a non-local separable potential, in terms of the partial-wave scatteringamplitude. Analytical expressions for some equilibrium thermodynamic properties, such asthe Helmholtz free energy and second virial coefficient, as well as some non-equilibriumthermophysical properties, such as collision cross sections and transport collision integrals,are described.

The solvable three-dimensional rank-two separable potential model: partial-wave scattering

In this paper, we study the ℓth partial-wave scattering for particles subjected to a non-local rank-two separable potential. Analytic expressions for the scattering amplitude, bound and resonance states, phase shifts and time delays are obtained and numerical illustrations are provided.

A solvable model of hydrogenic impurities in quantum dots

A solvable model is developed for electronic structure calculations of shallow hydrogenicimpurities in two-dimensional quantum dots. We replace the actual Coulomb interaction(local potential) between the electron and the hydrogenic impurity by a projective operator(non-local separable potential) to determine the resulting electronic states in closed form. Itis shown that non-local separable potentials may be used to accurately calculate the energyshift of the electronic levels as a function of the size of the quantum dot and the impurityposition.

Scattering of Dirac particles from nonlocal separable potentials: The eigenchannel approach

An application of the new formulation of the eigenchannel method [R. Szmytkowski, Ann. Phys. (N.Y.) 311, 503 (2004)] to quantum scattering of Dirac particles from nonlocal separable potentials is presented. Eigenchannel vectors, related directly to eigenchannels, are defined as eigenvectors of a certain weighted eigenvalue problem. Moreover, negative cotangents of eigenphase-shifts are introduced as eigenvalues of that spectral problem. Eigenchannel spinor as well as bispinor harmonics are expressed throughout the eigenchannel vectors. Finally, the expressions for the bispinor as well as matrix scattering amplitudes and total cross section are derived in terms of eigenchannels and eigenphase-shifts. An illustrative example is also provided.