Bound States for a Klein–Gordon Particle in Vector Plus Scalar Generalized Hulthén Potentials in D Dimensions

The Green’s function associated with a Klein–Gordon particle moving in a D-dimensional space under the action of vector plus scalar q-deformed Hulthén potentials is constructed by path integration for \({q \geq 1}\) and \({\frac{1}{\alpha} \ln q < r < \infty}\). An appropriate approximation of the centrifugal potential term and the technique of space-time transformation are used to reduce the path integral for the generalized Hulthén potentials into a path integral for q-deformed Rosen–Morse potential. Explicit path integration leads to the radial Green’s function for any l state in closed form. The energy spectrum and the correctly normalized wave functions, for a state of orbital quantum number \({l \geq 0}\), are obtained. Eventually, the vector q-deformed Hulthén potential and the Coulomb potentials in D dimensions are considered as special cases.     

Asymptotic Normalization Coefficients in a Potential Model Involving Forbidden States

It is shown that values obtained for asymptotic normalization coefficients by means of a potential fitted to experimental data on elastic scattering depend substantially on the presence and the number n of possible forbidden states in the fitted potential. The present analysis was performed within exactly solvable potential models for various nuclear systems and various potentials without and with allowance for Coulomb interaction. Various methods for changing the number n that are based on the use of various versions of the change in the parameters of the potential model were studied. A compact analytic expression for the asymptotic normalization coefficients was derived for the case of the Hulthén potential. Specifically, the d + α and α + ¹²C systems, which are of importance for astrophysics, were examined. It was concluded that an incorrect choice of n could lead to a substantial errors in determining the asymptotic normalization coefficients. From the results of our calculations, it also follows that, for systems with a low binding energy and, as a consequence, with a large value of the Coulomb parameter, the inclusion of the Coulomb interaction may radically change the asymptotic normalization coefficients, increasing them sharply.     

Generalized Coherent States for <Emphasis Type="Italic">q</Emphasis>-Deformed Hyperbolic Pöshel-Teller Potential

In this paper, we solve the Schrödinger equation for q-deformed hyperbolic Pöshel-Teller (PT) potential and we obtain the wave function and ladder operators for it. We show that these operators satisfy commutation relations of su(2) Lie algebra. Then we build the generalized coherent states for this q-deformed potential. We show that for the case q=1, we can obtain the same generalized coherent states for usual hyperbolic PT potential.     

Parameterization of the nuclear Hulthén potentials

Within the formalism of supersymmetry-inspired factorization method, a two-term nuclear Hulthén potential has been developed and parameterized to reproduce the nucleon–nucleon scattering phase shifts for P and D partial wave states.     

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.     

On solutions of the Schrödinger equation for some molecular potentials: wave function ansatz

Making an ansatz to the wave function, the exact solutions of the D-dimensional radial Schrödinger equation with some molecular potentials, such as pseudoharmonic and modified Kratzer, are obtained. Restrictions on the parameters of the given potential, δ and ν are also given, where η depends on a linear combination of the angular momentum quantum number ? and the spatial dimensions D and δ is a parameter in the ansatz to the wave function. On inserting D = 3, we find that the bound state eigensolutions recover their standard analytical forms in literature.     

Conformal Klein-Gordon Equations and Quasinormal Modes

Using conformal coordinates associated with conformal relativity—associated with de Sitter spacetime homeomorphic projection into Minkowski spacetime—we obtain a conformal Klein-Gordon partial differential equation, which is intimately related to the production of quasi-normal modes (QNMs) oscillations, in the context of electromagnetic and/or gravitational perturbations around, e.g., black holes. While QNMs arise as the solution of a wave-like equation with a Pöschl-Teller potential, here we deduce and analytically solve a conformal ‘radial’ d’Alembert-like equation, from which we derive QNMs formal solutions, in a proposed alternative to more completely describe QNMs. As a by-product we show that this ‘radial’ equation can be identified with a Schrödinger-like equation in which the potential is exactly the second Pöschl-Teller potential, and it can shed some new light on the investigations concerning QNMs.     

An improved approximation scheme for the centrifugal term and the Hulthén potential

We present a new approximation scheme for the centrifugal term to solve the Schrödinger equation with the Hulthén potential for any arbitrary l -state by means of a mathematical Nikiforov-Uvarov (NU) method. We obtain the bound-state energy eigenvalues and the normalized corresponding eigenfunctions expressed in terms of the Jacobi polynomials or hypergeometric functions for a particle exposed to this potential field. Our numerical results of the energy eigenvalues are found to be in high agreement with those results obtained by using the program based on a numerical integration procedure. The s -wave (l = 0analytic solution for the binding energies and eigenfunctions of a particle are also calculated. The physical meaning of the approximate analytical solution is discussed. The present approximation scheme is systematic and accurate.     

Modeling of confinement potential for cylindrical quantum dot

Quantum states and energy levels of an electron in a cylindrical quantum dot with different models of confinement potentials are studied. Two models of confinement potentials, Morse potential and modified Pöschl-Teller potential, are considered. It is shown that due to distinction between symmetric and asymmetric nature of potentials, there is a fundamental difference in behavior of the ground levels of charge carriers in these potentials. At small values of the width of Morse potential, quantum emission of electron occurs which is not observed in case of the modified Pöschl-Teller potential.     

Spin Symmetry for Dirac Equation with the Trigonometric Pöschl-Teller Potential

Within the framework of the Dirac theory, the relativistic bound states for the trigonometric Pöschl-Teller (PT) potential are obtained in the case of spin symmetry. It is found from the numerical results that there exist only positive energy states for bound states in the case of spin symmetry. Also, the energy levels approach a constant when the potential parameter α goes to zero. The special case for equally scalar and vector trigonometric PT potential is also studied briefly.     

A modification of Einstein–Schrödinger theory that contains Einstein–Maxwell–Yang–Mills theory

The Lambda-renormalized Einstein–Schrödinger theory is a modification of the original Einstein–Schrödinger theory in which a cosmological constant term is added to the Lagrangian, and it has been shown to closely approximate Einstein– Maxwell theory. Here we generalize this theory to non-Abelian fields by letting the fields be composed of d × d Hermitian matrices. The resulting theory incorporates the U(1) and SU(d) gauge terms of Einstein–Maxwell–Yang–Mills theory, and is invariant under U(1) and SU(d) gauge transformations. The special case where symmetric fields are multiples of the identity matrix closely approximates Einstein–Maxwell–Yang–Mills theory in that the extra terms in the field equations are < 10^?13 of the usual terms for worst-case fields accessible to measurement. The theory contains a symmetric metric and Hermitian vector potential, and is easily coupled to the additional fields of Weinberg–Salam theory or flipped SU(5) GUT theory. We also consider the case where symmetric fields have small traceless parts, and show how this suggests a possible dark matter candidate.     

Constraints on <Emphasis Type="Italic">SU</Emphasis>(2) ⊗ <Emphasis Type="Italic">SU</Emphasis>(2) invariant polynomials for a pair of entangled qubits

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices \(\mathfrak{P}_ +\). Since elements of \(\mathfrak{P}_ +\) are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, \(\mathfrak{P}_ + \in \mathbb{R}^{15}\). We define \(\mathfrak{P}_ +\) explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.     

An Alternative Model of the Damped Harmonic Oscillator Under the Influence of External Force

In this paper we introduce the modified time-dependent damped harmonic oscillator. An exact solution of the wave function for both Schrödinger picture and coherent state representation are given. The linear and quadratic invariants are also discussed and the corresponding eigenvalues and eigenfunctions are calculated. The Hamiltonian is transformed to SU(1,1) Lie algebra and an application to the generalized coherent state is discussed. It has been shown that when the system is under critical damping case the maximum squeezing is observed in the first quadrature F _x . However, for the overcritical damping case the maximum squeezing occurs in the second quadrature F _y . Also it has been shown that the system for both cases is sensitive to the variation in the coherent state phase.     

Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n ? 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.     

Approximate analytical solutions of the effective mass Dirac equation for the generalized Hulthén potential with any <Emphasis Type="Italic">κ</Emphasis>-value

The Dirac equation, with position-dependent mass, is solved approximately for the generalized Hulthén potential with any spin-orbit quantum number κ. Solutions are obtained by using an appropriate coordinate transformation, reducing the effective mass Dirac equation to a Schrödinger-like differential equation. The Nikiforov-Uvarov method is used in the calculations to obtain energy eigenvalues and the corresponding wave functions. Numerical results are compared with those given in the literature. Analytical results are also obtained for the case of constant mass and the results are in good agreement with the literature.     

The Coulomb-oscillator relation on <Emphasis Type="Italic">n</Emphasis>-dimensional spheres and hyperboloids

We establish a relation between Coulomb and oscillator systems on n-dimensional spheres and hyperboloids for n≥2. We show that, as in Euclidean space, the quasiradial equation for the (n+1)-dimensional Coulomb problem coincides with the 2n-dimensional quasiradial oscillator equation on spheres and hyperboloids. Using the solution of the Schrödinger equation for the oscillator system, we construct the energy spectrum and wave functions for the Coulomb problem.     

Hulthén potential models for <Emphasis Type="BoldItalic">α−α</Emphasis> and <Emphasis Type="BoldItalic">α−</Emphasis>He<Superscript><Emphasis Type="BoldItalic">3</Emphasis></Superscript> elastic scattering

Simple Hulthén-type potential models are proposed to treat the α?α and \(\alpha {-} \text {He}^{3}\) elastic scattering. The merit of our approach is examined by computing elastic scattering phases through the judicious use of the phase function method. Reasonable agreements in scattering phase shifts are obtained with the standard data.     

Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)

Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction Φ on a lattice L?? ^d determines a lattice spin system with potential A _Φ . The pressure P(A _Φ ) and free energy F _AΦ (β)=?(1/β)P(βA _Φ ) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, (microscopic) potentials are determined by their (macroscopic) free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a Hölder continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question ``Can you hear the shape of a drum?''.     

The trinification model <Emphasis Type="Italic">SU</Emphasis>(3)<Superscript>3</Superscript> from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)³ model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)³ spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)³.