We show that one of the models proposed by Ahmed [Few-Body Syst. 66, 28 (2025)] does not support bound states and the eigenfunctions derived by the author are not square integrable. His second example supports bound states but the author failed to truncate the infinite series and consequently he derived functions that are not solutions to the eigenvalue equation. Consequently, the energies reported by Ahmed are in both cases unphysical and of no utility whatsoever.
We present a systematic comparison of the hierarchical fractional N-body framework of Chishtie (Phys. Open 27, 100391, [4]) with three classical benchmarks in gravitational few-body dynamics, establishing structural analogies with the Kepler two-body radial free-fall, Euler’s two-fixed-centre problem, and Hill’s restricted lunar theory, together with quantitative computational benchmarks. The framework derives exact closed-form solutions to hierarchical configurations through Riesz fractional dynamics, yielding the fundamental scaling law α _k = 2 - 2/(N_k+1) connecting the correlation exponent to subsystem particle number. For the hierarchical three-body configuration with N_k= 2 , the effective radial ODE is r̈ = -(2/N_k)ω ^2 r^(1-N_k)/N_k and the parametric solution r_3(θ ) = a^4sin ^4θ conserves energy at the double-precision floor ( std/E_0 ∼ 10^-16 ) with Chebyshev cubic inversion residuals below 3× 10^-15 . The effective potential satisfies the virial relation ⟨ T⟩ /⟨ V⟩ = 1/(2N_k) exactly over the finite infall interval, providing an independent consistency check on the framework. The parametric structure is closely analogous to that of the Kepler radial free-fall via eccentric anomaly representation and to the Euler two-fixed-centre problem at equal masses and large separation ( r_3 ≫ c , quadrupole/monopole ratio 0.49% at the reference separation); the Hill radius arises naturally as the inner-subsystem scale at hierarchical level k. The universal near-collision velocity exponent γ = 1 is confirmed analytically for all N_k≥ 1 ; the fitted position exponent increases monotonically from 1.02 ( N_k=1 ) to 1.22 ( N_k=5 ), remaining within the theoretically predicted interval (1, 2) throughout. Computational benchmarks demonstrate speedups of 212–505 times over DOP853 adaptive integration across all tested configurations while maintaining energy conservation at std/E_0 ≤ 1.82× 10^-16 across 15 parameter combinations.
We introduce a non-Hermitian operator which has exponentially decreasing energy levels and study the Aharonov–Bohm effect for bound states. Further, we include the two-dimensional harmonic oscillator in this non-Hermitian quantum system and obtain real energy levels with exponentially decrease. We also show that the Aharonov–Bohm effect for bound states exists in this second case. In both cases, we go further by showing that non-null persistent currents at zero temperature and non-null revival times exist.
In this paper, we study the discretization of free scalar quantum field theory using the Haar wavelet basis. The field operators are expanded in an orthonormal multi-resolution system and the expansion is truncated at finite resolution, with this application a finite-dimensional representation of the Hamiltonian is obtained. These procedures turn the operator in a form of quadratic Hamiltonian that describes a set of coupled harmonic oscillators. This construction provides an explicit realization of ultraviolet and infrared cutoffs within a wavelet framework also the basis functions preserve their orthogonality and spatial localization. The Haar wavelet basis has a property of compact support which allows a transparent implementation of the discretization procedure. Using this property we can create a matrix formulation suitable for numerical diagonalization. We analyze the structure of the discretized Hamiltonian and comment on how locality and symmetry considerations are affected by the Haar approximation, and identify the spectral implications of the nearest-neighbor coupling structure. We use the results to show how multiresolution techniques can be used as a constructive discretization scheme in quantum field theory and clarify both the practical advantages and intrinsic limitations of the Haar basis in this context.
Our previous papers provided a classical description of one-electron diatomic molecules and of charge exchange. In the present paper we advanced these studies by providing a classical description of two-electron diatomic molecules, consisting of the two nuclei of charges Z and Z' ≥ Z and two electrons. In particular, in laboratory and astrophysical plasmas, in the process of charge exchange between multicharged ions, there can form transient two-electron diatomic quasi-molecules containing nuclei of Z' ≥ Z>> 1 . We analyzed classical two-electron diatomic molecules in two configurations: one where the two electrons rotate in the same circular orbit, being at the opposite ends of the diameter; another where the two electrons rotate in two different quasi-circular orbits. We demonstrated that in the first configuration, the two-electron system is characterized by a lower energy in all cases than the corresponding one-electron system. Besides, we showed that the equilibrium range of the projection of the electrons orbit on the internuclear axis is smaller than in the corresponding one-electron case. We also analyzed the stability of the system against small fluctuations of the parameters of the electrons orbit and gave the ranges of the stability. Most importantly, for the first configuration we described classically the double charge exchange in plasmas. The feasibility of the classical description of the double charge exchange represents a counterintuitive result. For the second configuration, we demonstrated that it is stable when one electron is relatively close to one of the nuclei while the other electron is relatively close to the other nucleus.
Abstract Heavy quark symmetries are useful for predicting the existence of heavy states, their masses, and spin states. Despite numerous studies on the $$P_{c\bar{c}}(4440)$$ P c c ¯ ( 4440 ) and $$P_{c\bar{c}}(4457)$$ P c c ¯ ( 4457 ) heavy states, their spin states have not been previously determined. In this study, heavy symmetries are applied to predict the spin states. If the $$P_{c\bar{c}}(4440)$$ P c c ¯ ( 4440 ) and $$P_{c\bar{c}}(4457)$$ P c c ¯ ( 4457 ) states are considered $$\bar{D}^*\Sigma _c$$ D ¯ ∗ Σ c molecules, they can be classified as heavy partners. This classification may help clarify their potential connections with heavy antiquark-diquark symmetry partners. By utilizing these alternative spin assignments and the concept of heavy antiquark-diquark symmetry, it may be possible to estimate $$\Xi _{cc}^{(*)}\Sigma _c^{(*)}$$ Ξ cc ( ∗ ) Σ c ( ∗ ) states, and ultimately, their spin states, which have not been elucidated in experiments. In addition to these symmetries, the relationship between $$P_{c\bar{c}}$$ P c c ¯ and $$P_{c\bar{c}s}$$ P c c ¯ s pentaquarks can be constructed which supports the prediction of possible $$P_{c\bar{c}s}$$ P c c ¯ s states. The predicted masses of the $$P_{c\bar{c}s}(4338)$$ P c c ¯ s ( 4338 ) and $$P_{c\bar{c}s}(4459)$$ P c c ¯ s ( 4459 ) states align with several studies, allowing us to eliminate a specific spin state. One spin state appears to be favored, suggesting that $$P_{c\bar{c}}(4440)$$ P c c ¯ ( 4440 ) has $$J^P=\frac{1}{2}$$ J P = 1 2 and $$P_{c\bar{c}}(4457)$$ P c c ¯ ( 4457 ) has $$J^P=\frac{3}{2}$$ J P = 3 2 .
The precise determination of fragmentation functions (FFs) of hadrons relies on the accurate description of the differential cross sections obtained from both experimental high-energy hadron colliders and theoretical predictions at higher orders in quantum chromodynamics. Various phenomenological strategies have been employed to extract FFs. In this work, we analyze the use of kinematical cuts for reactions including pions and kaons in proton-antiproton collisions to isolate individual FF contributions. This study examines the feasibility of using a similar approach as in proton-proton colliders to analyze FF flavour separation [1]. In particular, we study photon-hadron production at colliders, including NLO QCD and LO QED corrections to reconstruct the partonic momentum fractions.
In this study, we analyze the nature of the $T_{c\bar{c}1}(3900)$ state using a uniformization approach, with a particular focus on different pole configurations. The $T_{c\bar{c}1}(3900)$ was observed in the $J/\psi \pi^\pm$ invariant mass spectrum near the $D\bar{D}^*$ threshold, suggesting its possible interpretation as a $D\bar{D}^*$ hadronic molecule. We investigate this structure by modeling the coupled-channel interaction between $J/\psi \pi$ and $D\bar{D}^*$ using a fitting function where the pole-based interaction is embedded. With this, we are able to generate arbitrary pole structures and extract physical insight from the resulting line shapes. These synthetic line shapes are then used to generate a training dataset for a machine learning model. Since the signal appears above the $D\bar{D}^*$ threshold, our analysis primarily focuses on pole configurations with at least one pole on the third Reimann Sheet. Once the machine has been trained on a synthetic dataset and has demonstrated good generalization capabilities, an inference will be done on the BESIII data set. With this method, our technique was able to infer that the signal observed by the BESIII collaboration has a pole-shadow pair configuration, which can be interpreted as compact tetraquark state.
The precise determination of fragmentation functions (FFs) of hadrons relies on the accurate description of the differential cross sections obtained from both experimental high-energy hadron colliders and theoretical predictions at higher orders in quantum chromodynamics. Various phenomenological strategies have been employed to extract FFs. In this work, we analyze the use of kinematical cuts for reactions including pions and kaons in proton-antiproton collisions to isolate individual FF contributions. This study examines the feasibility of using a similar approach as in proton-proton colliders to analyze FF flavour separation [1]. In particular, we study photon-hadron production at colliders, including NLO QCD and LO QED corrections to reconstruct the partonic momentum fractions.
One of the most important methods for comprehending the fundamental dynamics of high energy nuclear interactions is the investigation of multiparticle correlation. In this study two- and three-particle short-range pseudorapidity ( η ) correlations in ^84Kr -Em interaction at 1 A GeV are investigated and also examine whether they are energy and mass independent. The analysis is performed using two particle normalized correlation functions ( R_2 ) and three particle normalised correlation function ( R_3 ) in order to examine the underlying dynamics of multiparticle production and the possible clustering behavior of emitted particles. Cumulative shower formation is responsible for the high two-particle short-range η correlation seen in the reverse direction in the target rest frame. Additionally, an analysis of the cumulative η gap distribution is conducted, and the associated cluster features are presented.
In this study, we analyze the nature of the Tcc & strns;1(3900)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{c\bar{c}1}(3900)$$\end{document} state using a uniformization approach, with a particular focus on different pole configurations. The Tcc & strns;1(3900)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{c\bar{c}1}(3900)$$\end{document} was observed in the J/psi pi +/-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi \pi <^>\pm $$\end{document} invariant mass spectrum near the DD & strns;& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\bar{D}<^>*$$\end{document} threshold, suggesting its possible interpretation as a DD & strns;& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\bar{D}<^>*$$\end{document} hadronic molecule. We investigate this structure by modeling the coupled-channel interaction between J/psi pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi \pi $$\end{document} and DD & strns;& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\bar{D}<^>*$$\end{document} using a fitting function where the pole-based interpretation is embedded. With this, we are able to generate arbitrary pole structures and extract physical insight from the resulting line shapes. These synthetic line shapes are then used to generate a training dataset for a machine learning model. Since the signal appears above the DD & strns;& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\bar{D}<^>*$$\end{document} threshold, our analysis primarily focuses on pole configurations with at least one pole on the third Riemann Sheet. Once the machine has been trained on a synthetic dataset and has demonstrated good generalization capabilities, an inference will be done on the BESIII dataset. With this method, our technique was able to infer that the signal observed by the BESIII collaboration has a pole-shadow pair configuration, which implies the presence of a large non-molecular component.
This paper presents an analytical study of the one-dimensional Dunkl-Schrödinger equation with a modified Eckart potential. The analysis is carried out within the framework of Dunkl differential operators, which incorporate the Wigner parameter μ _1 . The classical Eckart potential is generalized by using four real parameters, a_1 , a_2 , a_3 , and a_4 , thereby providing greater flexibility in modeling short-range interactions. The resulting quantization condition yields an implicit expression for the energy spectrum E_n and the corresponding wavefunction ϕ _1(x) . Numerical computations were performed to examine the influence of each parameter on the obtained results. Moreover, several particular cases corresponding to specific potentials were investigated. For each case, analytical expressions for the energy spectrum and the wavefunctions were derived both in the Dunkl framework and in the classical limit μ _1→ 0 . The paper concludes with a concise summary of the main analytical findings.
In this paper, we study approximate bound state solutions of the Schrödinger equation for the new proposed non-central exponential potential as the sum of a five-parameter exponential-type potential and a ring-shaped potential, by applying the approach of supersymmetric quantum mechanics (SUSYQM) in the framework of the improved approximation scheme to the centrifugal potential. The Schrödinger equation with this model potential is separated into angular and radial components. The bound state energy eigenvalues for both the radial and angular parts of a non-relativistic equation with a new non-central model potential are derived, and the corresponding radial and angular wave functions are expressed in terms of Jacobi polynomials. We also discuss some important special cases from our model. Our results are consistent with previous studies in the literature. We also present the numerical results for the energy spectrum of some special cases of the diatomic molecules CO, NO, N _2 and HCL.
We investigate the geometric structure of two-neutron halo nuclei from the perspective of Efimov physics. Using the analytic three-body wave function obtained from the Faddeev equations in the unitary limit, we explore the connection between Efimov universality and the spatial configuration of these weakly bound systems. The internal geometry is quantified through probability densities, root-mean-square interparticle distances and characteristic opening angles, evaluated for different neutron–core mass ratios. Our results reveal a universal trend in the geometry of s-wave dominated halo nuclei, reflecting the universal correlations characteristic of the Efimov-like regime.
The two-center Coulomb scattering problem is revisited. It is shown that the scattering wavefunction can be interpreted in terms of distorted single-center Coulomb scattering solutions. The two-center properties are shown to manifest themselves merely in a non-integer quasi-angular momentum and an infinite series of short-range potentials decaying faster than r^-2 . Employing this distorted-wave ansatz, two-center Coulomb scattering phase shifts are determined numerically at the challenging low-energy regime with unprecedented accuracy.
In this study, we analyze the nature of the T_cc̅1(3900) state using a uniformization approach, with a particular focus on different pole configurations. The T_cc̅1(3900) was observed in the J/ψπ ^± invariant mass spectrum near the DD̅^* threshold, suggesting its possible interpretation as a DD̅^* hadronic molecule. We investigate this structure by modeling the coupled-channel interaction between J/ψπ and DD̅^* using a fitting function where the pole-based interpretation is embedded. With this, we are able to generate arbitrary pole structures and extract physical insight from the resulting line shapes. These synthetic line shapes are then used to generate a training dataset for a machine learning model. Since the signal appears above the DD̅^* threshold, our analysis primarily focuses on pole configurations with at least one pole on the third Riemann Sheet. Once the machine has been trained on a synthetic dataset and has demonstrated good generalization capabilities, an inference will be done on the BESIII dataset. With this method, our technique was able to infer that the signal observed by the BESIII collaboration has a pole-shadow pair configuration, which implies the presence of a large non-molecular component.
