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Semiclassical methods are essential in analyzing quantum mechanical systems. Although they generally produce approximate results, relatively rare potentials exist for which these methods are exact. Such intriguing potentials serve as crucial test cases for semiclassical approximations. Recent research has demonstrated a deep connection between supersymmetric quantum mechanics and the exactness of semiclassical methods. Specifically, the mathematical form of conventional shape-invariant potentials guarantees exactness in several related situations. In this paper, we review these recent results and discuss their significance.

As a simple example of the "blurring" influence of noncommutative geometry, a noncommutative version of a high-frequency perturbation of a background metric is calculated. The relation between the principal null vectors of the wave and those of the induced symplectic structure is in this case particularly clear.

The cluster decay process is studied in the WKB approximation based on the unified fission model. The cluster is considered to be emitted by tunneling through a potential barrier taken as the sum of the Coulomb potential, the centrifugal potential and the modified Woods–Saxon (MWS) nuclear potential. The results of our calculations are compared to those obtained by other theoretical models as well as experimental data. It is shown that the unified fission model with the MWS nuclear potential can be successfully used to evaluate the cluster decay half-lives of heavy nuclei.

This paper shows that WKB wave function can be expressed in the form of an adiabatic expansion. To build a bridge between two widely invoked approximation schemes seems pedagogically instructive. Further, “cubic-WKB” method that has been devised in order to overcome the divergence problem of WKB can be also presented in the form of an adiabatic approximation: The adiabatic expansion of a wave function contains a certain parameter. When this parameter is adjusted so as to make the next order correction vanish approximately, the adiabatic wave function becomes equivalent to that of the “cubic-WKB”.

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qⁿ. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman–Gelca–LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear q-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.

In this paper, we have studied QNM modes and absorption cross-sections of Born–Infeld–de Sitter black holes. WKB approximation is employed to compute the QNM modes of massless scalar fields. We have also used null geodesics to compute quasinormal modes in the eikonal approximation. In the eikonal limit, QNMs of black holes are determined by the parameters of the circular null geodesics. Unstable circular null orbits are derived from the effective metric which is obeyed by light rays under the influence of a nonlinear electromagnetic field. Comparison is shown with the QNM of the linear electromagnetic counterpart, the Reissner–Nordström black hole. Furthermore, the null geodesics are employed to compute the absorption cross-sections in the high frequency limit via the sinc approximation.

A comparative study of the S-matrix and the WKB methods for the calculation of the half widths of alpha decay of super heavy elements is presented. The extent of the reliability of the WKB methods is demonstrated through simple illustrative examples. Detailed calculations have been carried out using the microscopic alpha-daughter potentials generated in the framework of the double-folding model using densities obtained in the relativistic mean field theories. We consider alpha-nucleus systems appearing in the decay chains of super heavy parent elements having A = 277, Z = 112 and A = 269, Z = 110. For negative and small positive log τ_1/2 values the results from both methods are similar even though the S-matrix results should be considered to be more accurate. However, when log τ_1/2 values are large and positive, the width associated with such state is infinitesimally small and hence calculation of such width by the S-matrix pole search method becomes a numerically difficult problem. We find that overall, the WKB method is reliable for the calculation of half lives of alpha decay from heavy nuclei.

The properties of proton resonances in the p and sd shells are studied with a single-particle potential generated by self-consistent-mean field calculations. Besides, we also propose to calculate the proton-nucleus interaction by a double-folding procedure with the density of the daughter nucleus represented by the result of the same mean-field model. Without raising any adjustable parameter, these calculations provide reasonable estimates for the widths of observed proton resonances in light nuclei. The mean-field-based calculation is particularly useful for the predications of the properties of resonances in light exotic nuclei and the direct proton capture reactions of astrophysics.

The alpha decay (AD) chains of the nuclei having $Z=118$, 119 and 120 have been investigated in terms of different theoretical models. Decay mode results that are presented in this study have been probed over the possible isotopes of the aforementioned nuclei. In the decay mode predictions, the formula of Bao et al. and the formula proposed by Soylu have been used to calculate the spontaneous fission (SF) half-lives. The AD half-lives have been computed by using the Denisov and Khuedenko, Royer, Horoi, the universal decay law (UDL), the Viola–Seaborg–Sobiczewski (VSS), the universal curve (UNIV) formulas and Wentzel–Kramers–Brillouin (WKB) approximation with Bohr–Sommerfeld quantization condition for the nuclei that have the measured experimental half-lives. Therefore, the rms values of the results of the related expressions and WKB method have been determined, in this way, AD half-life calculations of the $Z=118$, 119 and 120 nuclei have been performed. According to the obtained results, SF half-life values for Bao et al. and Soylu are quite different from one approach to another, the predictions on decay modes of the $Z=118$, 119 and 120 nuclei show differences. The decay modes produced by using different models used in this study would be important for the predictions of the future experimental investigations.

The predictive accuracy of the unified fission model based on a modified Woods–Saxon potential (UFMWS) is assessed by systematically calculating the decay half-lives of alpha emitters from $Z=52$ to $Z=118$. The computed results are compared with recent experimental data as well as with results obtained using different semi-empirical formulas. After confirming the performance of the UFMWS, we focus on the alpha decay of even-Z superheavy nuclei with $120\le Z\le 130$. The comparison between the experimental and the theoretical alpha-decay energies extracted from four different mass tables as well as between the experimental half-lives and those calculated by UFMWS using the theoretical values have shown that the WS4 mass model is the most accurate in the region of superheavy nuclei. The alpha decay half-lives of the nuclei of interest have then been computed with the UFMWS by inputting the WS4 decay energies. The study of the neutron number variation of decay half-lives allowed to identify regions of increased stability and to predict neutron magic numbers.

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of Bérenger. We prove that the Bérenger and closely related layers define well-posed transmission problems in great generality. When the Bérenger method or one of its close relatives is well-posed, perfect matching is proved. The proofs use the energy method, Fourier–Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the Bérenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in Bérenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of Bérenger's algorithm for Maxwell's equations, our favorite choice leads to a strongly well-posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence. Open problems are indicated throughout.

Recently a new caculational scheme for effective actions in radial background fields was developed. The effective action is expressed as an infinite sum of partial-wave contributions, using the rotational symmetry of the system. The sum becomes convergent after proper regularization and renormalization, but the rate of convergence is rather slow. We introduce a systematic way of accelerating the rate of convergence. This method is based on a radial WKB series in the angular momentum cut-off. We demonstrate the power of this scheme by applying it to the calculation of instanton determinant in QCD.