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Based on the one-dimensional quantum mechanics on (anti)-de Sitter background [W. S. Chung and H. Hassanabadi, Mod. Phys. Lett. A32, 26 (2107)], we discuss the Ramsauer–Townsend effect. We also formulate the WKB method for the quantum mechanics on (anti)-de Sitter background to discuss the energy level of the quantum harmonic oscillator and quantum bouncer.

In this paper, the Wentzel–Kramers–Brillouin (WKB) method is extended to be applicable for conformable Hamiltonian systems, where the concept of conformable operator with fractional order $\alpha$ is involved. The WKB approximation for the $\alpha$-wave function is derived for potentials which slowly vary in space. Some illustrative examples to demonstrate the method are presented. The quantities of the conformable form are found to be in exact agreement with the corresponding traditional quantities when $\alpha =1$.

In this paper, we systematically investigate the cluster decay half-lives of heavy nuclei using the Wentzel–Kramers–Brillouin method and Bohr–Sommerfeld quantization condition for the Morse molecular potential. We determine the optimal nuclear potential parameters that best fit the experimental data of known cluster decay half-lives. Subsequently, we theoretically calculate the cluster decay half-lives for the heavy nuclei lacking experimental decay half-lives. Our findings reveal that the Morse model is highly effective in explaining the decay half-lives of the heavy nuclei, particularly when utilizing two degrees of freedom. This approach produces results that match the accuracy of empirical models found in the existing literature. Theoretical determinations of cluster decay half-lives with optimally derived model parameters are expected to guide future experimental studies.

The Reissner–Nordström metric encompasses three distinct interpretations: first, as a charged static black hole in the Einstein–Maxwell theory; second, within the brane world model, where the charge is construed as a tidal parameter emanating from extra dimensions; and third, in the context of the Einstein-aether theory, wherein the charge is associated with the aether parameter. Considering these interpretations, we derive analytic expressions for the proper oscillation frequencies of black holes, known as quasinormal modes, for scalar, electromagnetic and Dirac perturbations. Employing the higher WKB technique and an expansion in terms of the inverse multipole number, our analytic formulas exhibit notable agreement with previously published numerical data and time-domain integration results. Additionally, we verify the correspondence between null geodesics and eikonal quasinormal modes for these cases.

By expanding in powers of the inverse multipole number, we derive compact analytic expressions for the quasinormal modes of scalar and electromagnetic perturbations around regular Simpson–Visser black holes. These analytic formulas offer sufficient accuracy for the first and higher multipoles, except when the black hole is in a near-extremal state. The correspondence between null geodesics and high-frequency quasinormal modes is confirmed for the case under consideration. As an application, we show that Simpson–Visser black holes respect the Hod’s bound on the damping rate of the fundamental quasinormal mode.

In this study, microscopic nucleon–nucleon Double Folding (DF) and phenomenological potentials have been used to investigate $\alpha {+}^{40}$Ca reaction observables at sub-barrier energies. In the calculations, semi-classical Wentzel–Kramers–Brillouin (WKB) approach has been used in order to obtain the cross-sections and reaction rates of $\alpha {+}^{40}$Ca. Besides WKB approximation, we have also utilized Talys code in order to get the comparative results and find out the method differences. To estimate the reaction rates, energy-dependent cross-sections and astrophysical S-factors of $\alpha$+${}^{40}$Ca have been used. Herewith, differences between models and potentials have been demonstrated using the reaction rate estimates.

The half-lives of the $\alpha$ decay for even–even nuclei having $Z=62–118$ (164 nuclei) have been calculated by using the Wentzel–Kramers–Brillouin (WKB) method together Bohr–Sommerfeld (BS) quantization condition for cosh potential in two-different forms with and without the isospin effects for six different parameter sets. By comparing the obtained results with the experimental values, the rms deviations have been computed. It is obtained that when the cosh potential is used in 2 different type forms, the rms values are also changed and the better one for obtaining the half-lives is cosh-2 form. Even if the isospin-dependent potential does not have so much effect on rms values in this sort of WKB with BS calculations with cosh potential forms, considering the isospin effect together the isospin-dependent nuclear diffuseness parameter have a strong influence on the rms values. New hindrance factor formula based on liquid drop model was used in order to show the influence of hindrance factor on the $\alpha$ decay half-lives. It is pointed out that this new function of the hindrance factor does not depend on the magic numbers, it contains fewer free parameters and using of this improves the calculations positively. New estimates by using present model have also been made for even–even nuclei having with $Z=118–120$ which do not have the measured $\alpha$ decay half-lives yet and they have been compared the results of the ImSahu model. The present results would provide useful information and knowledge for the explanation of half-lives of $\alpha$ decay of nuclei and future possible experimental studies.

$Q$-value and half-life of elements in alpha decay chain of ${}^{294}$117, ${}^{293}$117, ${}^{291}$116 and ${}^{290}$116 were calculated using the Nuclear potential generated by double folding procedure and using the WKB method treating the alpha decay as a tunneling problem. The nuclear potential was parameterized using Woods–Saxon potential. Using this approach, the $Q$-value and half-life of next heaviest element in the alpha decay chain of element ${}^{290}$116 is predicted. It is proposed to use this to predict the $Q$-value and half-life of other higher elements in different alpha decay chains.

We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed.

We give a justification of the perturbation theory for contributions of Morse critical points to the spectral decomposition in deformation quantization.

Black hole’s quasinormal frequencies are basically the complex numbers which provide information about the relaxation of perturbations and depend on the characteristics of the spacetime and types of perturbations. In this paper, we evaluate the spectrum of the quasinormal modes of Hayward black hole in Einstein–Gauss–Bonnet gravity, Hayward black hole in anti-de Sitter space (AdS) spacetime, and 4-dimensional black hole in Einstein–Lovelock gravity. By utilizing the 6th-order WKB resonance technique, we examine the quasinormal modes frequencies $\omega$ by shifting the charge parameter $Q$ (it is also identified with the cosmological constant), circular harmonic index $l$, and mass of scalar field $m$. We observe that 6th-order WKB method gives quite high accuracy when the multipole number $l$ is larger than the overtone $n$. We observe that real and imaginary components of the quasinormal modes are not linear functions similar to Reisnner–Nordström-AdS. For large values of charge, quasinormal ringing becomes slower to settle down to thermal equilibrium and hence the frequency of the oscillation becomes smaller.

By using the WKB method to solve the eigenvalue problem that results from a linear bifurcation analysis, we show that for a cylindrical shell of arbitrary thickness made of a neo-Hookean material which is subjected to an external hydrostatic pressure, the circular cross-section buckles into a non-circular shape at a value of μ₁ which depends on A₁/A₂ and the mode number, where A₁ and A₂ are the undeformed inner and outer radii, and μ₁ is the ratio of the deformed inner radius to A₁. We show that this problem is amenable to the WKB method based on the largeness of the mode number. Similar to the case of a Varga cylindrical tube [Sanjarani Pour, 2005], the dependence of μ₁ on A₁/A₂ has a boundary layer structure. Simple asymptotic expressions for the bifurcation condition are obtained. It has been shown that the asymptotic results obtained in the outer and inner regions agree with the numerical results. The two sets of results obtained for the eigenfunction with the aid of WKB and determinant methods are almost indistinguishable.

Propagation behavior of horizontally polarized shear waves (SH-waves) in a piezo-composite structure is discussed, using the Wentzel–Kramers–Brillouin (WKB) method. The considered model is made by combining the Functionally Graded Piezoelectric Material (FGPM) layer and piezoelectric orthotropic substrate. The linear form spring model is considered to delineate the imperfection of interface. Moreover, the material properties of FGPM layer are varying linearly along the thickness direction. Dispersion relation is obtained for both electrically open and short cases. Numerical example and graphical representation have been provided to illustrate the effect of different parameters on the phase velocity of SH-waves. As a special case, dispersion relation has been obtained when the boundary is perfect. Results are compared for different orthotropic materials to add more specific observations. Finally, the outcome of this study is validated by matching it with classical Love wave result. Observations will be helpful in optimization of Love wave sensors and Surface Acoustic Wave (SAW) devices.

By applying the Lie–Trotter operator splitting method and the idea of the WKB method, we have developed a simple, accurate and efficient analytical approximation for pricing the constant elasticity of variance (CEV) spread options. The derived option price formula bears a striking resemblance to Kirk’s formula of the Black–Scholes spread options. Illustrative numerical examples show that the proposed approximation is not only extremely fast and robust, but it is also remarkably accurate for typical volatilities and maturities of up to two years.

Some classes of exact solutions of eikonal equation has been found by using some decomposition method. There is also prsented its application to quantum mechanics.