Narrow Results

A new analog formula of the bivariate Mittag-Leffler function is presented in this work. We prove some relationships with known special functions, integral representations and recurrence relations. Also, we study some further interesting properties of the given function including the general transform and fractional calculus. Finally, a new integral operator is investigated.

This study focuses on investigating the relativistic quantum dynamics of a fermionic oscillator field in the background of a minimal surface, specifically a catenoid bridge. Our primary objective is to analyze the bound-state solutions of this system by solving the non-perturbative radial wave equation. To achieve this, we consider two distinct scenarios. First, we explore the case where the particle is located inside the catenoid bridge and search for solutions to the wave equation. Subsequently, we extend our analysis to scenarios where the particle is positioned at a significant distance from the origin, comparable to the size of the catenoid bridge. In this regime, we obtain solutions to the wave equation to understand the behavior of the fermionic oscillator field. By examining these solutions, we investigate the influence of both the oscillator coupling and the spacetime background on the system.

In this analysis, we analytically obtain the eigenvalue solutions of non-relativistic quantum particles interacting with Mie-type potential in a topological defect geometry. Afterward, we study the thermodynamic properties for the quantum system, such as the vibrational free energy, mean energy, entropy, and specific heat using the partition unction and analyze the effects of topological defect. Furthermore, we calculated the Shannon entropy for this quantum system, thus measuring the loss of information about the particle’s location. This measure of information reveals the influence of the topological parameter on the location of the particle and provides us with more detailed parameters for a possible experimental detection of these quantum particles.

The analytic properties of the Gaussian hypergeometric function is reviewed and applied to the development of a Fortran function code. The code developed can be used to evaluate the hypergeometric function on the whole complex plane with arbitrary complex parametric values. In the process of numerically verifying the code, a trigonometric identity involving the hypergeometric function listed in most of the mathematical handbooks is found to be incorrectly stated.

Firstly, the solvability of some quantum models like Eckart and Rosen–Morse II are explained on the basis of the shape invariance theory. Then, two generalized types of the Klauder–Perelomov and Gazeau–Klauder coherent states are calculated for the models. By means of calculating the Mandel parameter, it is shown that the weight distribution function of the first type coherent states obeys the Poissonian and super-Poissonian statistics, however, the weight distribution function of the second type coherent states obeys the Poissonian and sub-Poissonian statistics.

Using the associated Jacobi differential equation, we obtain exactly bound states of the generalization of Woods–Saxon potential with the negative energy levels based on the analytic approach. According to the supersymmetry approaches in quantum mechanics, we show that these bound states by four pairs of the first-order differential operators, represent four types of the laddering equations. Two types of these supersymmetry structures, suggest the derivation of algebraic solutions by two different approaches for the bound states.

Using the realization idea of simultaneous shape invariance with respect to two different parameters of the associated Legendre functions, the Hilbert space of spherical harmonics Y_{n m}(θ,φ) corresponding to the motion of a free particle on a sphere is split into a direct sum of infinite-dimensional Hilbert subspaces. It is shown that these Hilbert subspaces constitute irreducible representations for the Lie algebra u(1,1). Then by applying the lowering operator of the Lie algebra u(1,1), Barut–Girardello coherent states are constructed for the Hilbert subspaces consisting of Y_{m m}(θ,φ) and Y_{m+1 m}(θ,φ).

Using simultaneous shape invariance with respect to two different parameters, we introduce a pair of appropriate operators which realize shape invariance symmetry for the monomials on a half-axis. It leads to the derivation of rotational symmetry and dynamical symmetry group H₄ with infinite-fold degeneracy for the lowest Landau levels. This allows us to represent the Heisenberg–Lie algebra h₄ not only by the lowest Landau levels, but also by their corresponding standard coherent states.

For the needs of non-perturbative quantum theory, an upgraded concept of solvability is proposed. In a broader methodical context, the innovation involves Schrödinger equations which are piecewise analytic and piecewise solvable in terms of special (in our illustrative example, Whittaker) functions. In a practical implementation of our symbolic-manipulation-based approach, we work with a non-analyticity in the origin. A persuasive advantage is then found in the both-sidedness of our iterative localization of the energies.

In this paper, we study a relativistic quantum dynamics of spin-0 scalar particle interacts with scalar potential in the presence of a uniform magnetic field and quantum flux in background of Kaluza–Klein theory (KKT). We solve Klein–Gordon equation in the considered framework and analyze the relativistic analogue of the Aharonov–Bohm effect for bound states. We show that the energy levels depend on the global parameters characterizing the spacetime, scalar potential and the magnetic field which break their degeneracy.

In this paper, we investigate the behavior of a scalar bosonic field within the backdrop of Eddington-inspired Born–Infeld gravity with a global monopole configuration. Additionally, we consider the presence of both scalar and vector potentials. To introduce the scalar potential, we perform a transformation on the mass term, replacing $M^2$ with ${(M+S)}^2$. For the vector potential, we apply a minimal coupling scheme, transforming $p_{\mu }\to D_{\mu }\equiv (p_{\mu }-e{A}_{\mu })$ within the relativistic Klein–Gordon equation. We opt for a harmonic oscillator scenario where the scalar and vector potentials are equal, i.e. $S=V\propto x^2$. Through this setup, we analytically solve the wave equation using the confluent Heun equation form. Subsequently, we delve into the quantum system in the absence of any potential, determining the permissible values for bound-state relativistic energy levels and the scalar field’s wave function. Our analysis reveals that the energy levels and wave function are influenced by various parameters present in the eigenvalue expression.

The associated Legendre functions for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics .

In this paper, we present an exact solution of the Klein–Gordon equation in the framework of the fractional-dimensional space, in which the momentum and position operators satisfying the $R$-deformed Heisenberg algebras. Accordingly, three essential problems have been solved such as: the free Klein–Gordon equation, the Klein–Gordon equation with mixed scalar and vector linear potentials and with mixed scalar and vector inversely linear potentials of Coulomb-type. For all these considered cases, the expressions of the eigenfunctions are determined and expressed in terms of the special functions: the Bessel functions of the first kind for the free case, the biconfluent Heun functions for the second case and the confluent hypergeometric functions for the end case, and the corresponding eigenvalues are exactly obtained.

In this paper, we study the relativistic quantum motions of spin-zero scalar bosons confined by the quantum flux field in the presence of Coulomb-type scalar potential in the background of a topologically nontrivial 4D space–time. Afterwards, we introduce a Coulomb-like vector potential through a minimal substitution in the wave equation and determine the eigenvalue solutions of the quantum system analytically. In fact, it is shown there that the nontrivial topology of the geometry, Coulomb-types scalar and vector potential, and the quantum flux field influence the energy profile and wave function of the scalar bosons and get them modified. Also, the gravitational analogue of the Aharonov–Bohm effect is observed because the energy eigenvalue depends on the geometric quantum phase.

This research paper delves into the study of a nonrelativistic quantum system, considering the interplay of noninertial effects induced by a rotating frame and confinement by the Aharonov–Bohm (AB) flux field with potential in the backdrop of topological defects, specifically a screw dislocation. We first focus on the harmonic oscillator problem, incorporating an inverse-square repulsive potential. Notably, it becomes evident that the energy eigenvalues and wave functions are intricately influenced by multiple factors: the topological defect parameter $\beta$ (representing the screw dislocation), the presence of a rotating frame engaged in constant angular motion with speed $\mathrm{\Omega }$ and the external potential. Then we study the quantum behavior of nonrelativistic particles, engaging in interactions governed by an inverse-square potential, all while taking into account the effects of the rotating frame. In both scenarios, a significant observation is made: the quantum flux field’s existence brings about a shift in the energy spectrum. This phenomenon bears a resemblance to the electromagnetic Aharonov–Bohm effect.

In this research work, the relativistic quantum dynamics of oscillator field in wormhole background with a topological defect produced by a cosmic string is investigated. We consider an example of Morris–Thorne-type wormhole including a osmic string and derived the wave equation of the relativistic quantum oscillator. Through the Heun equation, we solve analytically the radial equation and obtain the ground state energy level $E_{1,m}$ and the radial wave function ${\psi }_{1,m}$ as particular cases. In fact, it is shown that the eigenvalue solution of the oscillator field is influenced by the topological defect of cosmic string and shifted the result. Furthermore, the wormhole throat radius also modifies the energy levels and the wave function of the relativistic quantum oscillator.

In this paper, our principal objective is to investigate the impact of disclination and throat radius of a three-dimensional traversable wormhole on quantum oscillator fields. Specifically, we focus on Perry–Mann-type wormhole with disclination while also considering the influence of rainbow gravity’s. We derive the radial equation of the relativistic Klein–Gordon oscillator within this wormhole background under the effects of gravity’s rainbow and the analytical eigenvalue solution is obtained using the confluent Heun function. In fact, we show that the behavior of the oscillator fields is significantly influenced not only by the presence of disclination and the throat radius but also by the parameter of rainbow gravity’s. We choose various such rainbow functions to present and analyze the eigenvalue solutions of the quantum oscillator fields.

In this work, we investigate the quantum motions of nonrelativistic particles interacting with a potential in the presence of the Aharonov–Bohm (AB) flux field within a topological defect geometry, for example space–time with a distortion of a vertical line into a vertical spiral. We begin by deriving the radial Schrödinger wave equation, incorporating an anharmonic oscillator potential, which is a superposition of a harmonic oscillator and an inverse square potential, along with a constant term. The eigenvalue solution is obtained through the confluent Heun equation focusing on the ground state energy level and the radial wave function for the radial mode $n=1$ as an example and analyze the results. Subsequently, we use these results to molecular potential models, considering pseudoharmonic and shifted pseudoharmonic potentials. The derived eigenvalue solutions provide insights into the behavior of particles within these potentials. Expanding our exploration, we study the quantum system featuring only an inverse square potential in the presence of the quantum flux field in the same geometry background. Employing the same procedure, we determine the ground state energy level and the radial wave function. Notably, our findings reveal that the eigenvalue solutions are significantly influenced by the topological defect characterized by the parameter $\beta$, and the quantum flux field ${\mathrm{\Phi }}_{\mathrm{\text{AB}}}$. This influence manifests as a shift in the energy spectrum, drawing parallels to the gravitational analog of the AB effect.

In this paper, we conduct a comprehensive exploration of the relativistic quantum dynamics of spin-0 scalar particles, as described by the Duffin–Kemmer–Petiau (DKP) equation, within the framework of a magnetic space-time. Our focus is on the Bonnor–Melvin–Lambda (BML) solution, a four-dimensional magnetic universe characterized by a magnetic field that varies with axial distance. To initiate this investigation, we derive the radial equation using a suitable wave function ansatz and subsequently employ special functions to solve it. Furthermore, we extend our analysis to include Duffin–Kemmer–Petiau oscillator fields within the same BML space-time background. We derive the corresponding radial equation and solve it using special functions. Significantly, our results show that the geometry’s topology and the cosmological constant (both are related to the magnetic field strength) influence the eigenvalue solution of spin-0 DKP fields and DKP-oscillator fields, leading to substantial modifications in the overall outcomes.

The purpose of this research is to analyze the general equations of double diffusive magneto-free convection in a rate type fluid presented in non-dimensional form, and apply to a moving heated vertical plate as in the boundary layer flow up, with existence of externally magnetic fields which are either moving or fixed consistent with the plate. Thermal transport phenomenon is discussed in the presence of constant concentration coupled with first-order chemical reaction with exponential heating. An innovative definition in power law (CF) and Mittag-Leffler (ABC) kernels form time fractional operators are implemented to hypothecate the constitutive mass, heat and momentum equations. The results based on special functions are obtained by using the technique of Laplace transformation to tackle the non-dimensional equations for velocity, mass and energy. The contribution of mass, thermal and mechanical components on the dynamics of fluid is presented and discussed independently. An interesting property regarding the behavior of the fluid velocity is found when the movement is observed in the magnetic intensity along with the plate. In that situation, the fluid velocity is not to be zero when it is far and away from the plate. Moreover, the heat transfer aspects, flow dynamics and their credence on the parameters are drawn out by graphical illustrations along with some special cases for the movement of the plate which are also studied.