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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.

Recently there has been progress on the computation of two- and three-point correlation functions with two "heavy" states via semiclassical methods. We extend this analysis to the case of AdS₅×T^{1, 1}, and examine the suggested procedure for the case of several simple string solutions. By making use of AdS/CFT duality, we derive the relevant correlation functions of operators belonging to the dual gauge theory.

An alternative approach to the quantum pentahedron is proposed and a representation on the phase–space is introduced. Some special cases are given explicitly and discussed.

Here, we comment on the paper “Numerical study of the SWKB condition of novel classes of exactly solvable systems.” [Y. Nasuda and N. Sawado, Mod. Phys. Lett. A 36, 2150025 (2021).] We show that it misrepresents our prior work [J. Bougie, A. Gangopadhyaya and C. Rasinariu, J. Phys. A: Math. Theor. 51, 375202 (2018)], and clarify this misunderstanding.

A new independent approach to the granularity of space is derived. The Bohr–Sommerfeld length spectrum is computed and discussed. Some values of the spectrum are given and compared with those found canonically elsewhere.

In this work, we calculate the tensor power spectrum and the tensor-to-scalar ratio $r$ within the frame of the Starobinsky inflationary model using the improved uniform approximation method and the third-order phase-integral method. We compare our results with those obtained with numerical integration and the slow-roll approximation to second-order. We have obtained consistent values of $r$ using the different approximations, and $r$ is inside the interval reported by observations.

The interaction between two intense fields with a closed three-level atomic system in the cascade configuration is considered. We derive the steady-state expressions for the field-induced susceptibilities which are valid for all field strengths for the two fields. In particular, we show that each field influences the steady-state susceptibility and absorption experienced by the other field.

We demonstrate that all gauge instantons in a $d=3+1$ Yang–Mills theory, with generic topological vacuum charge $K$, correspond to soliton solutions and kink scalar fields in $d=4+1$ space-time.

The Bohr–Sommerfeld volume spectrum of polyhedron with arbitrary number of faces is computed. Some spacial cases (the three, four and five valent node Hilbert space) are studied.