Related papers: The Coulomb problem on a 3-sphere and Heun polynom…
We study the Hydrogen atom as a quantum mechanical system with a Coulomb like potential, with a semiclassical approach based on an effective description of quantum mechanics. This treatment allows us to describe the quantum state of the…
This paper presents analytical solutions for eigenvalues and eigenfunctions of the Schr\"odinger equation in higher dimensions, incorporating the Dunkl operator. Two fundamental quantum mechanical problems are examined in their exact forms:…
In this paper, a quantum dot mathematical model based on a two-dimensional Schr\"odinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a…
A new approach to the finite-gap property for the Heun equation is constructed. The relationship between the finite-dimensional invariant space and the spectral curve is clarified. The monodromies are calculated and are expressed as…
We use special quadrature formulas for singular and hypersingular integral to numerically solve the Schr\"{o}dinger equation in momentum space with the linear confinement potential, Coulomb and Cornell potentials. It is shown that the…
In this paper we construct generalizations to spheres of the well known Levi-Civita, Kustaanheimo-Steifel and Hurwitz regularizing transformations in Euclidean spaces of dimensions 2, 3 and 5. The corresponding classical and quantum…
The bound state energy eigenvalues for the two-dimensional Kepler problem are found to be degenerate. This "accidental" degeneracy is due to the existence of a two-dimensional analogue of the quantum-mechanical Runge-Lenz vector.…
We consider the problem of three identical charged particles on a plane under a perpendicular magnetic field and interacting through Coulomb repulsion. This problem is treated within Taut's framework, in the limit of vanishing center of…
We study the motion of a particle in a 3-dimensional lattice in the presence of a Coulomb potential, but we demonstrate semiclassicaly that the trajectories will always remain in a plane which can be taken as a rectangular lattice. The…
The geometric potential in quantum mechanics has been attracted attention recently, providing a formalism to investigate the influence of curvature in the context of low-dimensional systems. In this paper, we study the consequences of a…
The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere S^n following variational and optimal control approaches. The relation between the Hamiltonian…
We transform the Schr\"odinger wave equation to a nine-parameter Heun-type differential equation. Using our solutions of the latter in [J. Math. Phys. 59 (2018) 113507], we are able to identify the associated potential function, energy…
We obtain a symmetric tridiagonal matrix representation of the Dirac-Coulomb operator in a suitable complete square integrable basis. Orthogonal polynomials techniques along with Darboux method are used to obtain the bound states energy…
We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence). Our technique applies…
We consider Fock's fundamental theory of the hydrogen atom in momentum space which allows a realization of the previously predicted rotation group of a three-dimensional (3D) sphere in four-dimensional (4D) space. We then modify Fock's…
Spectrum and eigenfunctions in the momentum representation for 1D Coulomb potential with deformed Heisenberg algebra leading to minimal length are found exactly. It is shown that correction due to the deformation is proportional to square…
We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature…
We show that there exist some intimate connections between three unconventional Schr\"odinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively.…
We propose the assumption of quantum mechanics on a discrete space and time, which implies the modification of mathematical expressions for some postulates of quantum mechanics. In particular we have a Hilbert space where the vectors are…
A generalisation of the odd Bernoulli polynomials related to the quantum Euler top is introduced and investigated. This is applied to compute the coefficients of the spectral polynomials for the classical Lam\'e operator.