Approximate Solutions to the Dirac Equation with Effective Mass for the Manning–Rosen Potential in N Dimensions

The solutions of the effective mass Dirac equation for the Manning–Rosen potential with the centrifugal term are studied approximately in N dimension. The relativistic energy spectrum and two-component spinor eigenfunctions are obtained by the asymptotic iteration method. We have also investigated eigenvalues of the effective mass Dirac–Manning–Rosen problem for α = 0  or  α = 1. In this case, the Manning–Rosen potential reduces to the Hulthen potential.     

Groundstate threshold in disordered anisotropic +/−J Ising models

In disordered anisotropic square +/− J Ising models SQ(p, q) at groundstates we investigate the pairs (p_c, q_c) of critical concentrations of antiferromagnetic bonds with concentrations p,q, respectively in orthogonal coordinate directions. We are led to p_c(q) ≈ π(q) with π(q) from the so-called adjoined problem. This approach is well supported by simulations for different values of q on the basis of minimal matchings of frustrated plaquettes. In particular, p_c(0) ≈ 0.21 from simulations and π(0) = 0.2113248 …, with the conjecture that p_c(0) = π(0). The concept of the adjoined problem is extended to d-dimensional (hyper-) cubic lattices. We hereby obtain for p_c,d especially in the sotropic case: p_c,3 ≈ 0.154, p_c,4 ≈ 0.170, p_c,5 ≈ 0.178, p_c,6 ≈ 0.182. Moreover, in analogy to SQ(p,q) we used the approach also for honeycomb Ising models HC(p,q,r) with no antiferromagnetic bonds in the third plaquette direction (r = 0).     

Ground-state properties of the antiferromagnetic potts model in an external field

The ground-state properties of the antiferromagnetic q-state Potts model in an external field are studied. At the upper critical field this model may be mapped onto a hard-core lattice gas with activity z = q ?1. This allows us to get some exact results for the triangular lattice on which the corresponding hard hexagon problem has been recently solved by Baxter.     

Large energy cumulants in the 2D Potts model and their effects in finite size analysis

We develop an ansatz for expressing the free energy of the two-dimensional q-states Potts model for q > 4 near its first order phase transition point. We notice that for the moderate values of q < 15, the energy profile at the phase transition is not expressible as a sum of gaussians. We discuss how this affects the traditional finite size analysis of this phase transition. In particular, the dominant length scale governing the finite size corrections turns out to be much (∼ 6 times) larger than the largest correlation length in the problem.     

The inverse problem for the third order equation uxxx + q(x)ux + r(x)u = −iζ3u

The spectral problem u_xxx + q(x)u_x + r(x)u = ?iξ³u is considered. A set of spectral data which is sufficient for the reconstruction of the potentials q(x) and r(x) is found and the problem of this reconstruction, this inverse problem solved.     

Weighted Graph Colorings

We study two weighted graph coloring problems, in which one assigns q colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting w that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial Ph(G,q,w) associated with this problem that generalizes the chromatic polynomial P(G,q). General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for Ph(G,q,w) for lattice strip graphs G with periodic longitudinal boundary conditions. The zeros of Ph(G,q,w) in the q and w planes and their accumulation sets in the limit of infinitely many vertices of G are analyzed. Finally, some related weighted graph coloring problems are mentioned.     

Thermodynamic consistency of the q-deformed Fermi-Dirac distribution in nonextensive thermostatics

The q-deformed statistics for fermions arising within the nonextensive thermostatistical formalism has been applied to the study of various quantum many-body systems recently. The aim of the present note is to point out some subtle difficulties presented by this approach in connection with the problem of thermodynamic consistency. Different possible ways to apply the q-deformed quantum distributions in a thermodynamically consistent way are considered.     

Inverse Scattering Problem for a Two Dimensional Random Potential

We study an inverse problem for the two-dimensional random Schrödinger equation (Δ + q + k ²)u = 0. The potential q(x) is assumed to be a Gaussian random function whose covariance operator is a classical pseudodifferential operator. We show that the backscattered field, obtained from a single realization of the random potential q, determines uniquely the principal symbol of the covariance operator of q. The analysis is carried out by combining harmonic and microlocal analysis with stochastic methods.     

Tsallis q-Stat and the Evidence of Long-Range Interactions in Soil Temperature Dynamics

The complexities in the variations of soil temperature and thermal diffusion poses a physical problem that requires more understanding. The quest for a better understanding of the complexities of soil temperature variation has prompted the study of the q-statistics in the soil temperature variation with the view of understanding the underlying dynamics of the temperature variation and thermal diffusivity of the soil. In this work, the values of Tsallis stationary state q index known as q-stat were computed from soil temperature measured at different stations in Nigeria. The intrinsic variations of the soil temperature were derived from the soil temperature time series by detrending method to extract the influences of other types of variations from the atmosphere. The detrended soil temperature data sets were further analysed to fit the q-Gaussian model. Our results show that our datasets fit into the Tsallis Gaussian distributions with lower values of q-stat during rainy season and around the wet soil regions of Nigeria and the values of q-stat obtained for monthly data sets were mostly in the range 1.2≤q≤2.9 for all stations, with very few values q closer to 1.2 for a few stations in the wet season. The distributions obtained from the detrended soil temperature data were mostly found to belong to the class of asymmetric q-Gaussians. The ability of the soil temperature data sets to fit into q-Gaussians might be due and the non-extensive statistical nature of the system and (or) consequently due to the presence of superstatistics. The possible mechanisms responsible this behaviour was further discussed.     

Relativistic effects in one-dimensional hydrogen atom

The dependence of stationary levels of a Dirac particle in the regularized Coulomb potential V _??(z) = ?q/(|z| + ??) on the cutoff parameter ?? is studied. It is shown that, in 1 + 1 D, the energy spectrum of a Dirac particle in such a potential reveals some specific features which nonanalytically depend on the coupling constant q and are essentially relativistic in nature. These properties turn out to be most important for ?? ? 1, explicitly demonstrating the existence of a physically reasonable energy spectrum for an arbitrarily small ?? > 0 and, at the same time, the absence of regular limit ?? ?? 0 (hence, the absence of a well-defined spectral problem for the Dirac equation without regularization for arbitrary q in 1 + 1 D).     

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

We consider the scaling limit of the two-dimensional q-state Potts model for q ⩽ 4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one-and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit q → 1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.     

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.     

The Quantum Superalgebra {\mathfrak{osp}_{q}(1|2)} and a q-Generalization of the Bannai–Ito Polynomials

The Racah problem for the quantum superalgebra \({\mathfrak{osp}_{q}(1|2)}\) is considered. The intermediate Casimir operators are shown to realize a q-deformation of the Bannai–Ito algebra. The Racah coefficients of \({\mathfrak{osp}_q(1|2)}\) are calculated explicitly in terms of basic orthogonal polynomials that q-generalize the Bannai–Ito polynomials. The relation between these q-deformed Bannai–Ito polynomials and the q-Racah/Askey–Wilson polynomials is discussed.     

Comment on: “Any l-state solutions of the Klein-Gordon equation with the generalized Hulthén potential”

It is shown that the bound l-state solutions of the Klein-Gordon equation for the general scalar and vector Hulthén potentials obtained by Qiang et al. are valid only for q?1 and . We clarify the problem and give the correct solutions when 0<q<1 or q<0. In each case, we derive a transcendental quantization condition for the s-state energy levels.     

Frequency spectrum and the Mössbauer effect in thin films

The paper is devoted to the theoretical study of thermal vibrations and the Mössbauer effect in a cubic thin film. The frequency spectrum determined in the first neighbour approximation, was used to calculate the Debye-Waller factor, as a function of the film thickness, temperature and force constants. The main peculiarity of the phenomenon is the appearance in the recoiless fraction of an anisotropy with respect to the angleθ between the normal to the film surface and the direction of the emitted (or absorbed) gamma quanta. For a film withq monoatomic layers the Debye-Waller factor is given by the expression f(q)=f(∞)exp{c(q)[1+D)(q)cos² θ]}, where f(∞) is the bulk-body value. To obtain this compact analytical expression for f(q) the appearing sums overq modes of vibration were transformed into integrals, using the wellknown L. B.Euler's transformation relation. Then the mainq-dependence of c(q) andD(q) isc(q)=1+C ₁(T, λ)/q,D(q)=D ₁ (T, λ)/q whereλ denotes the ratio between transverse and longitudinal components of the force constant tensor. Although an analytic expression was obtained for the general case, an explicit temperature dependence was made obvious only for low and high temperatures. It was also pointed out that for a correct mathematical treatment of the problem the presence of the support with its surface conditions, was compulsory.     

Generation of exact analytic bound state solutions from solvable non-powerlaw potentials by a transformation method

A transformation method has been applied to the exactly solvable Hulthen problem to generate a hierarchy of exactly solved quantum systems in any chosen dimension. The generated quantum systems are, in general, energy-dependent with a single normalized eigenfunction, as the Hulthen potential is a non-powerlaw potential. A method has been devised to convert a subset of the generated quantum systems with energy-dependent potentials to a single normal system with an energy-independent potential that behaves like a potential qualitatively similar to the Poschl-Teller potential. A second-order application of the transformation method on the Hulthen system produces another Sturmian quantum system and a different method is given to regroup them into a normal quantum system which resembles the Morse potential. Existence of normalizable eigenfunctions for these systems are found to be dependent on the local and asymptotic behaviour of the transformation function. Received 30 August 2000 and Received in final form 16 March 2001     

Momentum distributions in the nucleus

The nuclear form factor F(q) and one particle momentum distribution p(q) can be shown to have a power law decrease for large momenta. For the form factor F(q) we show that it is q/A that must be large for this asymptotic behavior to be important. For only q large the form factor, in a simple model, is shown to decrease exponentially in q. A similar behavior for p(q) is proposed.     

Proving AGT conjecture as HS duality: Extension to five dimensions

We extend the proof from Mironov et al. (2011) [25], which interprets the AGT relation as the Hubbard-Stratonovich duality relation to the case of 5d gauge theories. This involves an additional q-deformation. Not surprisingly, the extension turns out to be straightforward: it is enough to substitute all relevant numbers by q-numbers in all the formulas, Dotsenko-Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β=1, i.e. for q=t in MacDonald?s notation. For β≠1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams.