Narrow Results

In the limit $\hslash \to 0$, we analyze a class of Schrödinger operators $H_{\hslash }={\hslash }^{2}L+\hslash W+V\cdot {\mathrm{\text{id}}}_{{ℰ}}$ acting on sections of a vector bundle ${ℰ}$ over a Riemannian manifold $M$ where $L$ is a Laplace type operator, $W$ is an endomorphism field and the potential energy $V$ has a non-degenerate minimum at some point $p\in M$. We construct quasimodes of WKB-type near $p$ for eigenfunctions associated with the low-lying eigenvalues of $H_{\hslash }$. These are obtained from eigenfunctions of the associated harmonic oscillator $H_{p,\hslash }$ at $p$, acting on smooth functions on the tangent space.

We examine in detail nonperturbative corrections for low lying energies of a symmetric triple-well potential with non-equivalent vacua, for which there have been disagreement about asymptotic formulas and controversy over the validity of the dilute gas approximation. We carry out investigations from various points of view, including not only a numerical comparison of the nonperturbative corrections with the exact values but also the prediction on the large order behavior of the perturbation series, consistency with the perturbative corrections, and comparison with the WKB approximation. We show that all the results support our formulas previously obtained from the valley method calculation beyond the dilute gas approximation.

The radiation of vector particles by black holes in (1+2) dimensions is investigated within the WKB approximation. We consider the process of quantum tunneling of bosons through an event horizon of the black hole. The emission temperature for the Schwarzschild background geometry coincides with the Hawking temperature and for the Rindler spacetime the temperature is the Unruh temperature. We also obtain the radiation temperature for the de Sitter spacetime.

We have extended the analytical transfer matrix (ATM) method to solve quantum mechanical bound state problems with complex PT-symmetric potentials. Our work focuses on a class of models studied by Bender and Jones, we calculate the energy eigenvalues, discuss the critical values of g and compare the results with those obtained from other methods such as exact numerical computation and WKB approximation method.

In this paper, we study the tunneling of charged fermions from the stationary axially symmetric black holes using the generalized uncertainty principle (GUP) via Wentzel, Kramers, and Brillouin (WKB) method. The emission rate of the charged fermions and corresponding modified Hawking temperature of Kerr–Newman black hole, Einstein–Maxwell-dilaton-axion (EMDA) black hole, Kaluza–Klein dilaton black hole, and then, charged rotating black string are obtained and we show that the corrected thermal spectrum is not purely thermal because of the minimal scale length which cause the black hole’s remnant.

In this paper, we study the quasinormal modes for scalar, electromagnetic, and gravitational axial perturbations in the Hayward regular black hole surrounded by quintessence (HBH-${\omega }_q$). Using the third-order WKB approximation, we can determine the dependence of the quasinormal modes on the parameters of the regular black hole and the parameters on the test fields. We also determine the greybody factor, giving transmission and reflection coefficients of the scattered wave through the effective potentials in the WKB approximation using numerical analysis.

The black hole entropy due to spin fields are calculated by using brick-wall model. The appearance of the logarithmic terms is demonstrated and we specially deal with the subleading logarithmic term which exists for any spin fields. It is shown that the subleading logarithmic term is related to the use of WKB approximation but it usually includes not only a quadratic term and a linear term of the spin but also a zero-power term of the spin.

In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

In this paper, we discuss the effects of the topology of a disclination on quantum revivals when a quantum particle is subject to an infinite cylindrical well and a logarithmic potential. By using the WKB (Wentzel, Kramers and Brillouin) approximation, we obtain the eigenvalues of energy and show that the influence of the topology of the disclination on them gives rise to an Aharonov–Bohm-type effect. Further, we discuss how the topology of a disclination influences the classical periods and the revival times.

Following our previous study of the Bohr–Sommerfeld (BS) quantization condition for one-dimensional case [J. C. del Valle and A. V. Turbiner, Int. J. Mod. Phys. A36, 2150221 (2021)], we extend it to $d$-dimensional power-like radial potentials. The BS quantization condition for $S$-states of the $d$-dimensional radial Schrödinger equation is proposed. Based on numerical results obtained for the spectra of power-like potentials, $V(r)=r^m$ with $m\in [-1,\infty )$, the correctness of the proposed BS quantization condition is established for various dimensions $d$. It is demonstrated that by introducing the WKB correction$\gamma$ into the right-hand side of the BS quantization condition leads to the so-called exact WKB quantization condition, which reproduces the exact energies, while $\gamma$ remains always very small. For $m=2$ (any integer $d$) and for $m=-1$ (at $d=2$) the WKB correction $\gamma =0$: for $S$ states the BS spectra coincides with the exact ones. Concrete calculations for physically important cases of linear, cubic, quartic and sextic oscillators, as well as Coulomb and logarithmic potentials in dimensions $d=2,3,6$ are presented. Radial quartic anharmonic oscillator is considered briefly.

The paper presents a complete description of the periodic solutions of the Duffing equation:

These results are obtained by showing that the half-period Poincaré map offered by the Duffing equation is asymptotically equivalent to a map of a circle into itself, according to:

Two systems of hyperbolic equations, arising in the multiphase semiclassical limit of the linear Schrödinger equations, are investigated. One stems from a Wigner measure analysis and uses a closure by the Delta functions, whereas the other relies on the classical WKB expansion and uses the Heaviside functions for closure. The two resulting moment systems are weakly and non-strictly hyperbolic respectively. They provide two different Eulerian methods able to reproduce superimposed signals with a finite number of phases. Analytical properties of these moment systems are investigated and compared. Efficient numerical discretizations and test-cases with increasing difficulty are presented.

We study the half-lives of some Pt isotopes via the alpha-decay process and a modified barrier penetration formula different from the traditional barrier potentials. Comparison with the existing data is quite motivating.

In this paper, the heavy cluster half-lives were calculated using the Wentzel–Kramers–Brillouin (WKB) method. First, we have considered the cluster and daughter nuclei as spherical and the half-lives have been obtained. Then, we have assumed two nuclei as deformed ones and calculations have been performed once again. Deformations of the emitted cluster and daughter nucleus have been presented in terms of elongation and neck thickness parameters. We investigated the influence of elongation and neck thickness parameters on cluster decay half-lives. Our results show that half-lives are affected by these parameters and obtained results for deformed nuclei are in better agreement with experimental data.

The alpha decay half-lives of even–even and even–odd Platinum (Pt) nuclei have been studied within the Coulomb and proximity potential model (CPPM). The present study is restricted to even–even nuclei with $A=166$–198. The results are compared with other calculations such as the Semi-empirical formula (SemFIS) from Poenaru et al. based on fission theory of alpha decay, the Viola–Seaborg (VS), Royer (R) and Brown formulae. Also, the alpha decay half-lives have been calculated using the Scaling law of Brown (SLB), the Universal Decay Law (UDL) of Qi et al., the Scaling Law of Horoi et al. (SLH), and Akrawy–Dorin formula (ADF) of Akrawy and Poenaru, which are the Royer modified formula for alpha decay half-live by adding asymmetry term.

A novel technique is presented to analytically solve the fractional diffusion equation for non-reactive air pollutants emitted from an elevated continuous source into the air. A generalized methodical solution combining first-order Wentzel–Kramers–Brillouin (WKB) approximation theory and the Sturm–Liouville problem is used to solve the air pollutants’ fractional dispersion equation. Drawing insight from previous analysis, we expanded the initial issue assuming the turbulent flow characteristics appearing in the diffusion process in a non-integer dimensional space. We solved the transformed problem and compared the solutions against data from real experiment. Physical consequences connecting to the conventional generalized diffusion equations are presented. The results indicate that the present solutions are in accordance with those obtained in literature. This report demonstrates that fractional equations can be applied in a practical prediction of pollutant dissemination in a turbulent atmosphere.

A general theory is given for quantizing both constrained and unconstrained systems with second-order Lagrangian, using the WKB approximation. In constrained systems, the constraints become conditions on the wave function to be satisfied in the semiclassical limit. This is illustrated with two examples.

We consider an elastic medium with a disclination and investigate the topological effects on the interaction of a spinless electron with radial electric fields through the WKB (Wentzel, Kramers, Brillouin) approximation. We show how the centrifugal term of the radial equation must be modified due to the influence of the topological defect in order that the WKB approximation can be valid. Then, we search for bound states solutions from the interaction of a spinless electron with the electric field produced by this linear distribution of electric charges. In addition, we search for bound states solutions from the interaction of a spinless electron with radial electric field produced by uniform electric charge distribution inside a long non-conductor cylinder.

We analyze the interaction of the induced electric dipole moment of a neutral particle with an electric field in elastic medium with a charged disclination from a semiclassical point of view. We show that the interaction of the induced electric dipole moment of a neutral particle with an electric field can yield an attractive inverse-square potential, where it is influenced by the topology of the disclination. Then, by using the Wentzel, Kramers and Brillouin approximation based on the Langer transformation, we show that the centrifugal term of the radial equation must be modified due to the influence of the topology of the disclination. Besides, we obtain the bound states solutions to the Schrödinger equation.

In this paper, we study the topological effects of the global monopole spacetime on the energy eigenvalues of spherical symmetric potentials in the nonrelativistic regime. We deal with the radial equation by using the Wentzel, Kramers and Brillouim (WKB) approximation. In the cases where the energy levels of the $\ell$-waves can be achieved, the WKB approximation is used based on the Langer transformation.