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This paper explores the Bohr Hamiltonian with a newly introduced generalized form of the sextic potential in the $\beta$-part of the nuclear potential. The generalized sextic potential proposed in this research is based on hyperbolic functions and is characterized by parameters a, b, c, d, and $\eta$. It is quite versatile and reduces to several well-known potential models of physical interest, making it particularly rich for nuclear structure studies. We employed the extended Nikiforov–Uvarov method and confluent Heun equation to obtain analytical solutions of the $\beta$-part of the Schrödinger equation for the Bohr Hamiltonian, with the Greene–Aldrich approximation scheme applied to the centrifugal barrier term. The results of calculations are employed to study the energy spectra and electric quadrupole transition rates of ${}^{128}$Xe, ${}^{130}$Xe, ${}^{132}$Xe, ${}^{134}$Xe, ${}^{192}$Pt, ${}^{194}$Pt and ${}^{196}$Pt atomic nuclei. Our model successfully reproduces experimental data of the isotopes under investigation, demonstrating good agreement in energy spectra and $B(E2)$ transition rates. Based on the significance of the parameter $\eta$, the generalized sextic potential provides a valuable tool for studying various unexplored potential models and transitions in nuclear structure, with potential applications in identifying new states of triaxial nuclei.

A basic theoretical model known as the Schrodinger equation is utilized to explain the physics of the waveform, a phenomenon in quantum mechanics. The whole characterization of a technique’s components is included within the wave function. There are several aspects of the Schrodinger equation’s numerical solution. In this work, we determined mathematically the eigenvalues and characteristic functions of the Eckart-Hellmann potential. This study adopted an estimating strategy for solving the problem recommended by the Nikiforov–Uvarov approach and employed the Greene–Aldrich approximation. The aim was determining bound energies and applying the findings to particular diatomic molecules and their spectroscopic parameters. By contrasting our eigenvalue data with extra numerical data collected by other scholars, the good outcomes of our technique were validated.

We study, by using numerical computation methods, the structure of the energy spectra of the quantum relativistic rotating oscillators we have recently proposed, starting with a set of one-parameter deformations of the static (3+1) de Sitter or anti-de Sitter metrics.¹

Approximative analytic solutions of the Dirac equation in the geometry of Schwarzschild black holes are derived obtaining information about the discrete energy levels and the asymptotic behavior of the energy eigenspinors.

In this paper, the wave equation corresponding to the $\gamma$-rigid version of Bohr Hamiltonian for the modified Davidson potential is investigated in the position-dependent mass formalism. By solving the related differential equation, the wave function, energy spectra and transition rates are obtained. In order to evaluate our results, they are compared with experimental data through the standard error.

Here, analytical expressions of energy eigenvalues and eigen functions for a generalized Cornell potential are obtained by solving the non-relativistic Schrodinger equation using the Nikiforov–Uvarov functional analysis method along with Greene–Aldrich approximation. Energy spectra of three physically important potentials viz the pseudoharmonic, the Kratzer and the Coulomb perturbed potentials are derived from the general results. Further, within the framework of the Kratzer potential, energy eigenvalue spectra of diatomic molecules $\mathrm{\text{CO}}$, $\mathrm{\text{NO}}$, ${\mathrm{\text{O}}}_2$, $\mathrm{\text{LiH}}$ and $\mathrm{\text{HCl}}$ are computed. The mass spectra of two heavy mesons are also investigated using the Coulomb perturbed potential, a form of the generalized Cornell potential. The obtained results are in good agreement with the results of others studies. The study is further extended to calculate and draw the partition function and other associated thermodynamic quantities for heavy mesons.

In the presented analysis of air shower data measured with the KASCADE experiment energy spectra for five mass groups are reconstructed. The results show a change of composition towards heavier elements across the knee but also demonstrate an insufficient description of the data by the used hadronic interaction models QGSJet and SIBYLL.

For one-dimensional power-like potentials $|x{|}^m$, $m>0$, the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for $m>2$ but below them for $m<2$. The ground state BSE as function of $m$ are of the same order of magnitude as the exact energies for linear $(m=1)$, quartic $(m=4)$ and sextic $(m=6)$ oscillators but their relative deviation grows with $m$, reaching the value 4 at $m=\infty$. For physically important cases $m=1,4,6$, for the 100th excited state BSE coincide with exact ones in 5–6 figures.

It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction$\gamma$ (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function $\left|\gamma \right|<1/2$ for all $m\ge 1$. It grows slowly with increasing $m$ for fixed quantum number $N$, while it decays with quantum number growth at fixed $m$. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of $10^{-9}\text{–}10^{-11}$ (and $10^{-6}$).

We calculate the quantum energy spectra of molecular crystals, such as acetanilide, by using discrete nonlinear Schrodinger equation, containing various interactions, appropriate to the systems. The energy spectra consist of many energy bands, in each energy band there are a lot of energy levels including some higher excited states. The result of energy spectrum is basically consistent with experimental values obtained by infrared absorption and Raman scattering in acetanilide and can also explain some experimental results obtained by Careri et al. Finally, we further discuss the influences of variously characteristic parameters on the energy spectra of the systems.

The extension of the quantization rule approach to non-central potentials is investigated. The energy spectra for the generalized Coulomb and oscillator systems are presented. The results are in good agreement with those obtained before.

The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.

We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose time-evolution is determined from the underlying Euler equations. We prove partial well-posedness results for dissipative statistical solutions and propose a Monte Carlo type algorithm, based on spectral viscosity spatial discretizations, to approximate them. Under verifiable hypotheses on the computations, we prove that the approximations converge to a statistical solution in a suitable topology. In particular, multi-point statistical quantities of interest converge on increasing resolution. We present several numerical experiments to illustrate the theory.

Rotational structures of even–even ^150–160Nd nuclei are studied with the self-consistent deformed Hartree–Fock (HF) and angular momentum (J) projection model. Spectra of ground band, recently observed K = 4^-, K = 5^- and a few more excited, positive and negative parity bands have been studied up to high spin values. Apart from these, detailed electromagnetic properties (like E2, M1 matrix elements) of all the bands have been obtained. There is substantial agreement between our model calculations and available experimental data. Predictions are made about the band structures and electromagnetic properties of these nuclei. Some 4-qasiparticle K-isomeric bands and their electromagnetic properties are predicted.

In this paper, by using the SO(6) representation of eigenstates and transitional Interacting Boson Model (IBM) Hamiltonian, the evolution from prolate to oblate shapes along the chain of Hg isotopes is studied. Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are found to be in good agreement with experimental data for ^200–204Hg isotopes which are supported to be located in this transitional region.

In this work, we investigate the $\gamma$-rigid version of Bohr–Hamiltonian for the modified Davidson potential. Since the corresponding wave equation cannot be solved analytically, we apply the variational method. The related wave function, energy spectra and transition rates are determined. In order to evaluate our results, we fit the formula for the energy spectra to the available experimental data for some nuclei and compare the obtained standard error with the corresponding one in other similar work. Moreover, we study the collective behavior of these nuclei through the evolution of two quantities $E(2_1)$ and $\frac{E(4_1)}{E(2_1)}$ in terms of number of valence nucleons.

The present work is aimed at considering the recent forms of Bohr Hamiltonian, which are namely the hybrid model and the model combining the X(3) and E(5) symmetries, in the presence of the $\beta$-dependent Morse potential. The energy spectra and the transition rates of each model have been obtained. Some nuclei, the isotopes of Ru, Pd, Xe and ${}^{134}$Ba, have been fitted by using the three-parameter solution of the combined Hamiltonian with the Morse potential. Also, a few nuclei have been fitted by using the four-parameter solution of the hybrid model. In order to evaluate our results, in addition to reporting the root mean square (rms), we compare our data for each nucleus with the corresponding results of other references.

The Bohr Hamiltonian with four inverse power terms potential for the $\beta$-part and a harmonic oscillator for the $\gamma$-part is solved. The $\beta$-part has been solved using the biconfluent Heun equation. The total wave function and energy have been derived. The numerical results for energy triaxial nuclei spectra are compared with the experimental data, esM and esKM models known for ${}^{192,194,196}\mathrm{\text{Pt}}$ atomic nuclei. These results are in overall good agreement with the experimental data. After this, the corresponding $B(E2)$ transition rates have been calculated for each nuclei of Platinum.

In this work, Bohr Hamiltonian is used to explain the behavior of triaxial nuclei. A new potential, called Morse plus screened Kratzer potential, has been developed for the $\beta$-part with $\gamma$ fixed at $\frac{\pi }{6}$. The Extended Nikiforov–Uvarov method involving Confluent Heun functions is used to derive the wave function and energy expression. The electric quadrupole transition rates and energy spectrum of platinum ${}^{192,194,196}\mathrm{\text{Pt}}$ are determined and compared with the experimental data and some theoretical results.

In this paper, we look for a novel analytical solution of the Conformable Fractional Bohr Hamiltonian in the presence of a four inverse power terms potential in $\beta$-part of the collective nuclear potential. The new expression for the wave functions and energy spectra is obtained using the conformable fractional extended Nikiforov–Uvarov method. The effect of the conformable fractional formulation is studied on root mean square deviation, and normalized fractional eigen energies of ground state band, $\gamma$-band and $\beta$-band of ${}^{192}$Pt, ${}^{194}$Pt and ${}^{196}$Pt atomic nuclei. The $B(E2)$ transition rates are calculated within the fractional domain. We compare our results with the experimental data, the classical Bohr Hamiltonian model with four inverse power terms potential, and other relevant theoretical works. We discuss on how the fractional parameter $\alpha$ acts on the spectra of triaxial nuclei. The results provided by our new approach are found to be in good agreement with the experimental data and improved compared to previous works.

In this paper, we look for a novel analytical solution of the Bohr Hamiltonian in the presence of the Hulthen plus screened Kratzer potential in $\beta$-part of the collective nuclear potential. The new expression for the energy spectra is obtained using the method of super symmetric quantum mechanics. The effect of the Hulthen plus screened Kratzer potential is studied on root mean square deviation, and normalized eigen energies of ground state band, $\gamma$-band and $\beta$-band of ¹⁹²Pt, ¹⁹⁴Pt and ¹⁹⁶Pt atomic nuclei. The $B(E2)$ transition rates are calculated for the latter potential. Furthermore, our results are compared with experimental data, and other relevant theoretical works. We discuss on how the combination of these two potentials makes it possible to improve the results of energy spectra and quadrupole transitions of triaxial nuclei. We discuss the quantitative and qualitative descriptions concerning the structure of the nucleus using some indicators such as root mean square and staggering of the $\gamma$-band. The results provided by our model are found to be in good agreement with experimental data and improved in comparison with previous works.