Infinitesimal Parameters Research Articles

A Novel Exactly Theoretical Solvable of Bound States of the Dirac-Kratzer-Fues Problem with Spin and Pseudo-Spin Symmetry

New exact bound state solutions of the deformed radial upper and lower components of Dirac equation and corresponding Hermitian anisotropic Hamiltonian operator are studied for the modified Kratzer-Fues potential (m.k.f.) potential by using Bopp’s shift method instead to solving deformed Dirac equation with star product. The corrections of energy eigenvalues are obtained by applying standard perturbation theory for interactions in one-electron atoms. Moreover, the obtained corrections of energies are depended on two infinitesimal parameters (θ,χ), which induced by position-position noncommutativity, in addition to the discreet nonrelativistic atomic quantum numbers: (j=l±1/1,s=±1/2,landm) and we have also shown that, the usual relativistic states in ordinary three dimensional spaces are canceled and has been replaced by new degenerated 2(2l+1) sub-states in the extended quantum symmetries (NC: 3D-RS).

Nonresonant two-photon transitions in length and velocity gauges

We reexamine the invariance of two-photon transition matrix elements and corresponding two-photon Rabi frequencies under the "gauge" transformation from the length to the velocity gauge. It is shown that gauge invariance, in the most general sense, only holds at exact resonance, for both one-color as well as two-color absorption. The arguments leading to this conclusion are supported by analytic calculations which express the matrix elements in terms of hypergeometric functions, and ramified by a "master identity" which is fulfilled by off-diagonal matrix elements of the Schroedinger propagator under a the transformation from the velocity to the length gauge. The study of the gauge dependence of atomic processes highlights subtle connections between the concept of asymptotic states, the gauge transformation of the wave function, and infinitesimal damping parameters for perturbations and interaction Hamiltonians that switch off the terms in the infinite past and future [of the form exp(-epsilon * | t |)]. We include a pertinent discussion.

A New Study to the Schrödinger Equation for Modified Potential <i>V(r)=ar<sup>2</sup>+br<sup>-4</sup>+cr<sup>-6</sup></i> in Nonrelativistic Three Dimensional Real Spaces and Phases

In present work, by applying Boopp’s shift method and standard perturbation theory we have generated exact nonrelativistic bound states solution for a modified potential (see formula in paper) in both three dimensional noncommutative space and phase (NC: 3D-RSP) at first order of two two infinitesimal parameters antisymmetric (see formula in paper), we have also derived the corresponding noncommutative Hamiltonian.

A New Study to the Schrödinger Equation for Modified Potential <i>V(r)=ar<sup>2</sup>+br<sup>-4</sup>+cr<sup>-6</sup></i> in Nonrelativistic Three Dimensional Real Spaces and Phases

In present work, by applying Boopp’s shift method and standard perturbation theory we have generated exact nonrelativistic bound states solution for a modified potential (see formula in paper) in both three dimensional noncommutative space and phase (NC: 3D-RSP) at first order of two two infinitesimal parameters antisymmetric (see formula in paper), we have also derived the corresponding noncommutative Hamiltonian.

Demonstration of the sensitivity of the Smith predictor to parameter uncertainties using stability diagrams

Time-domain representation of the original Smith predictor is presented for systems with feedback delay. Sensitivity to parameter uncertainties is analyzed for a marginally stable, for an asymptotically stable and for an unstable second-order plant. A series of stability diagrams are constructed using the D-subdivision method and Stepan’s formulas. Transition to the stability diagrams subjected to delayed state feedback is established. It is demonstrated that the Smith predictor is sensitive to infinitesimal parameter mismatches for the marginally stable plant. It is shown that the Smith predictor can stabilize unstable plants for some extremely detuned internal model parameters.

Holomorphic bundles and the moduli space of N=1 supersymmetric heterotic compactifications

We describe the first order moduli space of heterotic string theory compactifications which preserve N =1 supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on T X to a field, the moduli space corresponds to deformations of a holomorphic structure $$ \overline{D} $$ on a bundle $$ \mathcal{Q} $$ . The bundle $$ \mathcal{Q} $$ is constructed as an extension by the cotangent bundle T ∗ X of the bundle E = End(V )⊕End(TX)⊕TX with an extension class $$ \mathcal{H} $$ which precisely enforces the anomaly cancelation condition. The deformations corresponding to the bundle E are simultaneous deformations of the holomorphic structures on the poly-stable bundles V and T X together with those of the complex structure of X. We discuss the fact that the “moduli” corresponding to End(T X) cannot be physical, but are however needed in our mathematical structure to be able to enforce the anomaly cancelation condition. In the appendix we comment on the choice of connection on T X which has caused some confusion in the community before. It has been shown by Ivanov and others that this connection should also satisfy the instanton equations, and we give another proof of this fact.

Rotating Charged Hairy Black Hole in (2+1) Dimensions and Particle Acceleration

In this paper, we construct rotating charged hairy black hole in (2+1) dimensions for infinitesimal black hole charge and rotation parameters. Then we consider this black hole as particle accelerator and calculate the center-of-mass energy of two colliding test particles near the rotating charged hairy black hole in (2+1) dimensions. As we expected, the center-of-mass energy has infinite value.

Extended Chaplygin gas equation of state with bulk and shear viscosities

In this note extended Chaplygin gas equation of state includes bulk and shear viscosities suggested. Bulk viscosity assumed as power law form of density and shear viscosity considered as a constant. We study evolution of dark energy density numerically for several forms of scale factor, and analytically under some assumptions corresponding to early universe. We found our model is stable for infinitesimal viscous parameters.

Aspects of finite field-dependent symmetry in SU(2) Cho–Faddeev–Niemi decomposition

In this Letter we consider SU(2) Yang–Mills theory analyzed in Cho–Faddeev–Niemi variables which remains invariant under local gauge transformations. The BRST symmetries of this theory are generalized by making the infinitesimal parameter finite and field-dependent. Further, we show that under appropriate choices of finite and field-dependent parameter, the gauge-fixing and ghost terms corresponding to Landau as well as maximal Abelian gauge for such Cho–Faddeev–Niemi decomposed theory appear naturally within functional integral through Jacobian calculation.

Order-disorder quantum phase transition in the quasi-one-dimensional spin-1/2 collinear antiferromagnetic Heisenberg model

The ground-state properties of the quasi-one-dimensional spin-1/2 antiferromagnetic Heisenberg model is investigated by using a variational method. Spins on chains along the x direction are antiferromagnetically coupled with exchange J>0, while spins between chains in the y direction are coupled either ferromagnetically (J' < 0) or antiferromagnetically (J' > 0). The staggered and the colinear antiferromagnetic magnetizations are computed and their dependence on the anisotropy parameter λ=|J'|/J is analyzed. It is found that an infinitesimal interchain coupling parameter is sufficient to stabilize a long-range order with either a staggered magnetization m_{s} (J' > 0) or a colinear antiferromagnetic magnetization m_{caf} (J' < 0), both behaving as ≃λ¹/² for λ → 0.

Scattering Characteristics of 2-D Imperfect Cloaks With Layered Isotropic Materials

Transformation-based cloaks prescribing infinitesimal and infinite material parameters are difficult to be realized in practical applications. However, the underlying theory of electromagnetic transformation can be used in various applications, such as radar cross section (RCS) reduction. In this letter, we examine scattering characteristics of a metallic object coated with layered isotropic materials derived from transformation electromagnetics. The proposed structure can be viewed as an imperfect cloak, which, however, maintains the feature of reduced electromagnetic scattering while it can be easily realized in practice. Numerical experiments are performed to validate the idea.

Exciton Condensation Under High Magnetic Field

The new results in the theory of Bose-Einstein condensation (BEC) of the two-dimensional (2D) magnetoexcitons formed by the high-density electron-hole (e-h) pairs created on the semiconductor mono-layer in a strong perpendicular magnetic field are reviewed. One of them is the metastable dielectric liquid phase (MDLP) formed by the 2D magnetoexcitons BEG-ed on the single-particle state with sufficiently large values of the wave vector k, so that its product kl with the magnetic length l equals about kl approximate to 3-4. This state was revealed in the conditions when the electrons and holes are situated on the lowest Landau levels (LLLs) and the polarizability of the Bose gas was calculated on the base of the Anderson-type coherent excited states. They give rise to correlation energy and to chemical potential displaying a nonmonotonous dependence on the filling factor v(2) with a relative minimum and with positive compressibility in its vicinity. The influence of the excited Landau levels (ELLs) on the quantum states of the e-h system is due to the virtual quantum transitions of particles from the LLLs to ELLs during the Coulomb scattering processes and to their subsequent return back. These quantum transitions were taken into account in the frame of the second order perturbation theory giving rise to an effective Hamiltonian describing the supplementary indirect interactions between the particles lying on the LLLs. This interaction is characterized by a small parameter equal to the ratio r of the magnetoexciton ionization potential l(ex)(0) to the Landau quantization energy (h) over bar omega(c). The parameter r = l(ex)(0)/(h) over bar omega(c), decreases as H-1/2 with the increasing the magnetic field strength H. The supplementary interaction is attractive, making the magnetoexcitons in the Hartree approximation more robust. Nevertheless its exchange, Fock terms as well as the Bogoliubov u-v transformation terms give rise to positive, repulsion-type contributions to the chemical potential. The Bose gas of magnetoexcitons with k = 0 becomes weakly nonideal when the ELLs are taken into account. The collective elementary excitations of two ground states corresponding to BEG-ed magnetoexcitons forming either a nonideal Bose gas with k = 0 or the MDLP with kl approximate to 3-4 were studied in the frame of the perturbation theory with the infinitesimal parameter v(2)(1 - v(2)), chosen as a product of the filling factor v(2) and of the phase space filling factor (1 - v(2)). The collective elementary excitations in both cases consist from the exciton and plasmon branches. Due to the presence of the condensate there are energy and quasi-energy branches. The self-energy parts containing the unknown frequency in denominators increase the degree of the dispersion equations and give rise to mixed exciton-plasmon and exciton-exciton elementary excitation branches.

Structural Instability and Minimal Realizations

In this note, we consider the structural stability of an unstable plant, feedback stabilized by a controller. It is shown that infinitesimal parameter perturbations in the plant transfer matrix can destabilize the stable feedback control system obtained from a minimal realization of the ¿nominal plant.¿ The remedy for such structural instability consists in using a minimal realization in which the dimension of the ¿antistable part¿ is maximized over the uncertain parameter values. This requires knowledge of the ¿correct class¿ of parametrizations to which the nominal plant transfer matrix belongs.

Numerical Analysis of One-Dimensional Consolidation in Layered Clay Using Interface Boundary Relations in Terms of Infinitesimal Strain

The interface boundary relations are derived in this study for the numerical analysis of one-dimensional consolidation in multilayered clay profiles. The finite difference solutions are formulated based on Mikasa’s consolidation equation with infinitesimal strains and constant consolidation parameters under the same fundamental assumptions and limitations of the classic Terzaghi equation. Numerical examples are presented for multilayer clay profiles under single and double drainage conditions that validate the predicted excess pore pressures, strains, settlements, and rates of consolidation using interface boundary relations in terms of infinitesimal strains that are equivalent to those expressed in terms of excess pore pressures.

First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes

For a, b &gt; 0, we consider a temporally homogeneous, one-dimensional diffusion process X(t) defined over I = (-b, a), with infinitesimal parameters depending on the sign of X(t). We suppose that, when X(t) reaches the position 0, it is reflected rightward to δ with probability p &gt; 0 and leftward to -δ with probability 1 - p, where δ &gt; 0. Closed analytical expressions are found for the mean exit time from the interval (-b, a), and for the probability of exit through the right end a, in the limit δ → 0+, generalizing the results of Lefebvre, holding for asymmetric Wiener process. Moreover, in alternative to the heavy analytical calculations, a numerical method is presented to estimate approximately the quantities above. Furthermore, on the analogy of skew Brownian motion, the notion of skew diffusion process is introduced. Some examples and numerical results are also reported.

Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space

For a perturbed mechanical system in phase space, considering d̃/dt in the structure equation and process of proof including infinitesimal parameter ∊ obviously, this paper studies the perturbation to Mei symmetry and adiabatic invariants. Firstly, the exact invariant induced directly from the Mei symmetry of the system without perturbation is given. Secondly, based on the concept of high-order adiabatic invariant, the determining equations of the perturbation to Mei symmetry are established, the condition of existence of the Mei adiabatic invariant led by the perturbation to Mei symmetry is obtained, and its form is presented. Lastly, an example is given to illustrate the application of the results.

Rotational-vibrational energy spectra of triatomic molecules near relative equilibria

Complete Hamiltonian operators have been obtained in terms of internal coordinates on the basis of the fiber bundle theory in geometry [J. Math Phys. 44, 4411 (2003)]. In this article, the full Hamiltonian is specialized for a rigid and for a semirigid molecule. For the rigid molecule, all internal coordinates are fixed at constants, so that the Hamiltonian operator comes to take an ordinary matrix form, and accordingly, the Schrödinger equation becomes an algebraic eigenvalue equation. The eigenvalues then provide rotational energy spectra of the rigid molecule. For the semirigid molecule, the full Hamiltonian is expanded in the vicinity of an equilibrium position into a power series in an infinitesimal parameter, to which the perturbation method is applied to obtain energy spectra in the form of a power series in the infinitesimal parameter. Indeed, the energy spectra are calculated to the second order term in the infinitesimal parameter in both the cases where the unperturbed energy spectra are nondegenerate and where those are degenerate. It then turns out that the amount of the energy shift caused by the rotation, which is the sum of the pure rotational and the rotation-vibration coupling terms, is proportional to the total angular momentum eigenvalue. It is also observed that a vibrational energy shift occurs simultaneously, which is caused by the metric defined on the internal space. No symmetry is assumed on the shape of the molecule in equilibrium.

Approximation in Thin Film Brick-Wall Model

The thin film brick-wall model is an important method on calculating the entropy of the non-static black hole. But we can see in this paper that the approximation used in the thin film brick-wall model is a rate of two infinitesimal parameters in fact. in this paper, we make an exact calculation of the approximation term and give the solution in the Schwarzschild black hole space-time.

First Passage Problems for Asymmetric Wiener Processes

The problem of computing the moment generating function of the first passage time T to a &gt; 0 or −b &lt; 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

First Passage Problems for Asymmetric Wiener Processes

The problem of computing the moment generating function of the first passage time T to a &amp;gt; 0 or −b &amp;lt; 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.