Energy-dependent Potential Research Articles

Propagating-wave approximation in two-dimensional potential scattering

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $\psi(\mathbf{r})$ of the Schr\"odinger equation, $(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has the remarkable property that their Fourier transform $\tilde\psi(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

Unitarization of infinite-range forces: graviton-graviton scattering

A method to unitarize the scattering amplitude produced by infinite-range forces is developed and applied to Born terms. In order to apply S-matrix techniques, based on unitarity and analyticity, we first derive an S-matrix free of infrared divergences. This is achieved by removing a divergent phase factor due to the interactions mediated by the massless particles in the crossed channels, a procedure that is related to previous formalisms to treat infrared divergences. We apply this method in detail by unitarizing the Born terms for graviton-graviton scattering in pure gravity and we find a scalar graviton-graviton resonance with vacuum quantum numbers (JPC = 0++) that we call the graviball. Remarkably, this resonance is located below the Planck mass but deep in the complex s-plane (with s the usual Mandelstam variable), so that its effects along the physical real s axis peak for values significantly lower than this scale. This implies that the corrections to the leading-order amplitude in the gravitational effective field theory are larger than expected from naive dimensional analysis for s around and above the peak position. We argue that the position and width of the graviball are reduced when including extra light fields in the theory. This could lead to phenomenological consequences in scenarios of quantum gravity with a large number of such fields or, in general, with a low-energy ultraviolet completion. We also apply this formalism to two non-relativistic potentials with exact known solutions for the scattering amplitudes: Coulomb scattering and an energy-dependent potential obtained from the Coulomb one with a zero at threshold. This latter case shares the same J = 0 partial-wave projected Born term as the graviton-graviton case, except for a global factor. We find that the relevant resonance structure of these examples is reproduced by our methods, which represents a strong indication of their robustness.

Application of quantum supersymmetry to rovibrational states of diatomic molecules with an energy dependent Morse potential

Application of quantum supersymmetry to rovibrational states of diatomic molecules with an energy dependent Morse potential

Charge-current generations and optical specifications of Gaussian quantum dot with energy-dependent potential

We consider the energy-dependent quantum dot with GaAs/GaAlAs Gaussian potential encompassment. Depending on the experimental or theoretical results for different types of interactions, the energy dependency can also be chosen as quadratic, fractional or any other type. The solutions of the wave equation for the energy-dependent Gaussian quantum dot are performed numerically by employing the Runge-Kutta-Fehlberg method. The optical specifications of the structure are examined by using the compact-density-matrix formalism within iterative method. The total refractive index changes, the total absorption coefficients, the charge-currents and the induced magnetic fields are studied for different values of the energy dependency parameter.

Integration of the Kaup–Boussinesq system with a self-consistent source via inverse scattering method

In this study we consider the Kaup–Boussinesq system with a self-consistent source. We show that the Kaup–Boussinesq system with a self-consistent source can be integrated by the method of inverse scattering theory. For a solving the problem under consideration, we use the direct and inverse scattering problem of the Sturm–Liouville equation with an energy-dependent potential. The time evolution of the scattering data for the Sturm–Liouville equation with an energy-dependent potentials associated with the solution of the Kaup–Boussinesq system with a self-consistent source is determined. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Kaup–Boussinesq system with a self-consistent source.

Collective motion in γ-unstable nuclei within energy-dependent Davidson potential and deformation dependent mass formalisms

In this work, we propose an exactly solvable model which is constructed by considering energy-dependent Davidson potential in the β part of the generalized version of the collective quadrupole Bohr Hamiltonian (BH) within deformation-dependent mass (DDM) formalism. Analytical expression of the energy spectra and corresponding wave functions are derived by means of the asymptotic iteration method. The combined effect of DDM and the energy dependence of the potential coupling constant is duly investigated. Also, the numerical calculations of the electric quadrupole transition ratios and energy spectrum of several nuclei undergoing a γ-unstable shape phase transition are performed and compared with experimental data as well as with other theoretical models. Besides, we investigate the correlation between both formalisms: energy-dependent potential and DDM, through solutions of BH for transition nuclei in the limit E(5) with Davidson potential.

Analysis of coupled-channel potentials with quark and hadron degrees of freedom

The quark-antiquark potentials are known to be confining in the absence of the q̅q pair creation. On the other hand, the inter-hadron potentials vanish at large distance, because the interaction range is limited by the inverse pion mass. When the q̅q pair creation and annihilation are switched on, the channel coupling occurs between the quark and hadron degrees of freedom, modifying the behavior of potentials. In this work, we investigate the properties of the effective potentials obtained by eliminating one of the channels. We show that the coupling to the eliminated channel induces a non-local and energy dependent effective potential, irrespective of the properties of the transition potential. In addition, when the hadron channel having continuous scattering eigenstates is eliminated, the resulting inter-quark potential contains an imaginary part above the threshold to describe the decay effect into the meson pair.

Confinement in bilayer graphene via intra- and inter-layer interactions

We consider confinement of Dirac fermions in AB-stacked bilayer graphene by inhomogeneous on-site interactions, (pseudo-)magnetic field or inter-layer interaction. Working within the framework of four-band approximation, we focus on the systems where the stationary equation is reducible into two stationary equations with 2 × 2 Dirac-type Hamiltonians and auxiliary interactions. We show that the localized states are given in terms of solutions of an effective Schrödinger equation with energy-dependent potential. We consider several scenarios where bilayer graphene is subject to inhomogeneous (pseudo-)magnetic field, on-site interactions or inter-layer coupling. In explicit examples, we provide analytical solutions for the states localized by local fluctuations or periodicity defects of the interactions.

Survey of deep sub-barrier heavy-ion fusion hindrance phenomenon for positive and negative Q-value systems using the proximity-type potential

A systematic survey of the accurate measurements of heavy-ion fusion cross sections at extreme sub-barrier energies is performed using the coupled-channels (CC) theory that is based on the proximity formalism. This work theoretically explores the role of the surface energy coefficient and energy-dependent nucleus-nucleus proximity potential in the mechanism of the fusion hindrance of 14 typical colliding systems with negative -values, including 11B+197Au, 12C+198Pt, 16O+208Pb, 28Si+94Mo, 48Ca+96Zr, 28Si+64Ni, 58Ni+58Ni, 60Ni+89Y, 12C+204Pb, 36S+64Ni, 36S+90Zr, 40Ca+90Zr, 40Ca+40Ca, and 48Ca+48Ca, as well as five typical colliding systems with positive -values, including 12C+30Si, 24Mg+30Si, 28Si+30Si, 36S+48Ca, and 40Ca+48Ca. It is shown that the outcomes based on the proximity potential along with the above-mentioned physical effects achieve reasonable agreement with the experimentally observed data of the fusion cross sections , astrophysical factors, and logarithmic derivatives in the energy region far below the Coulomb barrier. A discussion is also presented on the performance of the present theoretical approach in reproducing the experimental fusion barrier distributions for different colliding systems.

A novel model for evaluating preload based on spectral problem in system with double-bolted connections

Accurate evaluation of preload in system with double-bolted connections is critical for the service performance of engineering structures or equipment. However, the theoretical framework for assessing preload is imperfect. An equivalent approach from the tightening process of double-bolted joints to second-order spectral problem of the energy-dependent potential is firstly established. This paper proposes a novel nonlinear model of bolt preload considering physical and geometric parameters by proving the completely integrability of systems used bolted connections. The experimental device for hmeasuring displacement and bolt preload was developed to verify the accuracy of proposed model. Results show that the maximum relative error is not more than 3%. Finally, the influences of elastic modulus, density, holes spacing, and height on preload are analyzed.

Characterization of Darboux transformations for quantum systems with quadratically energy-dependent potentials

We construct three classes of higher-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials by means of generalized Wronskian determinants. Particular even-order cases reduce to the Darboux transformation for conventional (energy-independent) potentials. Our construction is based on an adaptation of the results for coupled Korteweg–de Vries equations [N. V. Ustinov and S. B. Leble, J. Math. Phys. 34, 1421 (1993)].

Symmetry potentials and in-medium nucleon-nucleon cross sections within the Nambu–Jona-Lasinio model in relativistic impulse approximation

In the relativistic impulse approximation (RIA), we study symmetry potentials and in-medium nucleon-nucleon (NN) cross sections with the Nambu-Jona-Lasinio (NJL) model that features chiral symmetry. The chiral symmetry that plays a fundamental role in the nonperturbative physics in the strong interaction is anticipated to add restrictive effects on the symmetry potentials and in-medium NN cross sections. For comparison, we also perform the study with the usual relativistic mean-field (RMF) model. The numerical results with the NJL and RMF models are similar at saturation density and below, since a priori fit was made to saturation properties. With the increase of nuclear density, the chiral symmetry starts to be restored partially in the NJL model, resulting in the explicit fall of the scalar density. In a large energy span, the symmetry potential acquires a significant rise for the partial restoration of the chiral symmetry, compared to the one with the RMF model. It is found that the in-medium NN cross sections in the RIA with the NJL and RMF models both increase with the density in the energy region interested in this study, whereas those with the NJL model increase sharply as long as a clear chiral symmetry restoration takes place. The different tendency of observables in density can be transmitted to the different energy dependence in the RIA. The NJL model is shown to have characteristic energy-dependent symmetry potentials and NN cross sections beyond saturation point, apart from the RMF models.

The Solutions on One-Dimensional Dirac Oscillator with Energy-Dependent Potentials and Their Effects on the Shannon and Fisher Quantities of Quantum Information Theory

The Solutions on One-Dimensional Dirac Oscillator with Energy-Dependent Potentials and Their Effects on the Shannon and Fisher Quantities of Quantum Information Theory

Arbitrary l-Solutions of the Schrodinger Equation in Arbitrary Dimensions for the Energy Dependent Generalized Inverse Quadratic Yukawa Potential

Within the framework of Nikiforov-Uvarov method, we obtained an approximate solution of the Schrodinger equation for the Energy Dependent Generalized inverse quadratic Yukawa potential model. The bound state energy eigenvalues for were computed for various vibrational and rotational quantum numbers. Special cases were considered when the potential parameters were altered, resulting into Energy Dependent Kratzer and Kratzer potential, Energy Dependent Kratzer fues and Kratzer fues potential, Energy Dependent Inverse quadratic Yukawa and Inverse quadratic Yukawa Potential, Energy Dependent Yukawa (screened Coulomb) and Yukawa (screened Coulomb) potential, and Energy Dependent Coulomb and Coulomb potential, respectively. Their energy eigenvalues expressions and numerical computations agreed with the already existing literatures.

Energy-dependent one-dimensional potentials and scattering of relativistic particles

Scattering behaviors of spin-0 and spin-1/2 particles interacting with one-dimensional energy-dependent barrier-type potentials are investigated by considering the Klein–Gordon and Dirac equations. The central question addressed by this study is how the energy-dependent barrier-type (rectangular, step, Hulthn, Woods–Saxon) characteristic affects the transmission and reflection coefficients of the Klein–Gordon and Dirac particles. Moreover, Klein’s paradox in the presence of energy-dependent step potential for the spin-1/2 particles is pointed out.

Bound state solutions of the Klein–Gordon equation with energy-dependent potentials

In this paper, we investigate the exact bound state solution of the Klein–Gordon equation for an energy-dependent Coulomb-like vector plus scalar potential energies. To the best of our knowledge, this problem is examined in literature with a constant and position dependent mass functions. As a novelty, we assume a mass-function that depends on energy and position and revisit the problem with the following cases: First, we examine the case where the mixed vector and scalar potential energy possess equal magnitude and equal sign as well as an opposite sign. Then, we study pure scalar and pure vector cases. In each case, we derive an analytic expression of the energy spectrum by employing the asymptotic iteration method. We obtain a nontrivial relation among the tuning parameters which lead the examined problem to a constant mass one. Finally, we calculate the energy spectrum by the Secant method and show that the corresponding unnormalized wave functions satisfy the boundary conditions. We conclude the paper with a comparison of the calculated energy spectra versus tuning parameters.

Energy-Dependent Chemical Potentials of Light Hadrons and Quarks Based on Transverse Momentum Spectra and Yield Ratios of Negative to Positive Particles

We describe the transverse momentum spectra or transverse mass spectra of π ± , K ± , p , and p ¯ produced in central gold-gold (Au-Au), central lead-lead (Pb-Pb), and inelastic proton-proton (pp) collisions at different collision energies range from the AGS to LHC by using a two-component (in most cases) Erlang distribution in the framework of multisource thermal model. The fitting results are consistent with the experimental data, and the final-state yield ratios of negative to positive particles are obtained based on the normalization constants from the above describing the transverse momentum (or mass) spectra. The energy-dependent chemical potentials of light hadrons ( π , K , and p ) and quarks ( u , d , and s ) in central Au-Au, central Pb-Pb, and inelastic (pp) collisions are then extracted from the modified yield ratios in which the contributions of strong decay from high-mass resonance and weak decay from heavy flavor hadrons are removed. The study shows that most types of energy-dependent chemical potentials decrease with increase of collision energy over a range from the AGS to LHC. The curves of all types of energy-dependent chemical potentials, obtained from the fits of yield ratios vs. energy, have the maximum at about 3.510 GeV, which possibly is the critical energy of phase transition from a liquid-like hadron state to a gas-like quark state in the collision system. At the top RHIC and LHC, all types of chemical potentials become small and tend to zero at very high energy, which confirms that the high energy collision system possibly changes completely from the liquid-like hadron-dominant state to the gas-like quark-dominant state and the partonic interactions possibly play a dominant role at the LHC.

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.

Construction of an equivalent energy-dependent potential by a Taylor series expansion

To construct a phase-equivalent energy-dependent local potential corresponding to a sum of local and nonlocal interactions, we use a simple method for expanding the wave function in a Taylor series up to the third order. We apply the constructed potentials to calculate the scattering phase shifts using the phase equation. The results for scattering in nucleon–nucleon, $$\alpha$$ –nucleon, and $$\alpha$$ – $$\alpha$$ systems agree reasonably well with the standard data.

Construction of an equivalent energy-dependent potential by a Taylor series expansion

Для построения фазово-эквивалентного зависящего от энергии локального потенциала, соответствующего сумме локального и нелокального взаимодействий, используется простой метод разложения волновой функции в ряд Тейлора вплоть до третьего порядка. Построенные потенциалы применяются для вычисления сдвигов фазы рассеяния с использованием соответствующего фазового уравнения. Результаты для рассеяния в системах нуклон-нуклон, $\alpha$-нуклон и $\alpha$-$\alpha$ находятся в разумном согласии со стандартными данными.