An Improvement of the Asymptotic Iteration Method for Exactly Solvable Eigenvalue Problems

We derive a formula that simplifies the original asymptotic iteration method formulation to find the energy eigenvalues for the analytically solvable cases. We then show that there is a connection between the asymptotic iteration and the Nikiforo-Uvarov methods, which both solve the second order linear ordinary differential equations analytically.     

Asymptotic Statistics of Zeroes for the Lamé Ensemble

The Lamé polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable (QCI) system of commuting, second-order differential operators P ₀=Δ, P ₁,…,P ^{N −1} acting on C ^∞(? ^N ). These operators naturally depend on parameters and thus constitute an ensemble. In this paper, we compute the limiting level-spacings distributions for the zeroes of the Lamé polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters. Received: 17 January 2001 / Accepted: 14 May 2001     

Baecklund Transformations for the Initial Problems of Nizhnich and Nizhnich－Novikov－Veselov Equations

The homogeneous balance method is a method for solving genera/partial differential equations (PDEs). In this paper we solve a kind of initial problems of the PDEs by using the special Baecklund transformations of the initial problem. The basic Fourier transformation method and some variable-separation skill are used as auxiliaries. Two initial problems of Nizlmich and the Nizlanich-Novikov-Veselov equations are solved by using this approach.     

Relativistic oscillator strengths for the Cs isoelectronic sequence and collapse of ⨍ and d orbitals

Model potential calculations have been performed to investigate the influence of valence-core electron exchange on oscillator strengths in the Cs isoelectronic sequence. It is demonstrated that inclusion of valence-core electron exchange in the model potential calculations of oscillator strengths for transitions involving orbitals which undergo the collapse phenomenon (usually d or f orbitals) is important and may modify the character of the transition (emission or absorption) due to the change in the relative positions of the states involved. In the collapse region, oscillator strengths for these transitions may also depend strongly on the form of the exchange approximation used. The reason for this influence is that valence-core electron exchange greatly facilitates the collapse phenomenon, thus affecting the positions of states and radial transition integrals. For transitions which do not involve orbitals undergoing collapse or occur in members of the isoelectronic sequence distant from the collapse region, the exchange effects, although still important are much weaker and good results may be obtained not only with a model potential employing empirically adjusted exchange but also for a simple, free from adjustable parameters, semiclassical exchange approximation.     

Lee–Yang Problems and the Geometry of Multivariate Polynomials

We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by Pólya–Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee–Yang type problems in statistical mechanics, combinatorics, and geometric function theory.     

One-Dimensional Relativistic Problems on the Bound States and Degradation for a Superposition of Two δ Potentials

For a superposition of two potentials, exact one-dimensional wave functions of the bound states with quantization conditions and exact wave functions of the degradation state with transmission and reflection coefficients have been found. The phenomenon of complete reflection taking place for potentials of this type is investigated.     

Dynamical Localization for the Random Dimer Schrödinger Operator

We study the one-dimensional random dimer model, with Hamiltonian H _ω =Δ+V _ω , where for all x∈ $\mathbb{Z}$ , V _ω(2x)=V _ω(2x+1) and where the V _ω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V>0. We show that, for all values of Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/ $\sqrt 2$ , the Lyapunov exponent vanishes only at the two critical energies given by E=±V. For the particular value V=1/ $\sqrt 2$ , respectively, V= $\sqrt 2$ , we show the existence of new additional critical energies at E=±3/ $\sqrt 2$ , respectively, E=0. On any compact interval Inot containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈ $\ell$ ²( $\mathbb{Z}$ ) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here $\psi _t = e^{- iH_{\omega ^t}} \psi$ , and P _I(H _ω) is the spectral projector of H _ωonto the interval I. In particular, if V>1 and V≠ $\sqrt 2$ , these results hold on the entire spectrum [so that one can take I=σ(H _ω)].     

Contributions of γd → πNN + π0 d channels to the spin asymmetry and the Gerasimov-Drell-Hearn integral for the deuteron

Contributions from the semi-exclusive channels γd → π^± NN + π⁰ d and γd → π⁰ X (X=pn or d) to the deuteron spin asymmetry and the Gerasimov-Drell-Hearn (GDH) integral are explicitly evaluated using an enhanced elementary pion photoproduction operator and a realistic, high-precision potential model for the deuteron wave function. The sensitivity of the results to the elementary pion photoproduction operator is also investigated and considerable dependence is found. Results for the deuteron GDH integral are compared with the measurements from A2 and GDH@MAMI Collaborations.     

Finite-Rank Multivariate-Basis Expansions of the Resolvent Operator as a Means of Solving the Multivariable Lippmann–Schwinger Equation for Two-Particle Scattering

Finite-rank expansions of the two-body resolvent operator are explored as a tool for calculating the full three-dimensional two-body T-matrix without invoking the partial-wave decomposition. The separable expansions of the full resolvent that follow from finite-rank approximations of the free resolvent are employed in the Low equation to calculate the T-matrix elements. The finite-rank expansions of the free resolvent are generated via projections onto certain finite-dimensional approximation subspaces. Types of operator approximations considered include one-sided projections (right or left projections), tensor-product (or outer) projection and inner projection. Boolean combination of projections is explored as a means of going beyond tensor-product projection. Two types of multivariate basis functions are employed to construct the finite-dimensional approximation spaces and their projectors: (i) Tensor-product bases built from univariate local piecewise polynomials, and (ii) multivariate radial functions. Various combinations of approximation schemes and expansion bases are applied to the nucleon-nucleon scattering employing a model two-nucleon potential. The inner-projection approximation to the free resolvent is found to exhibit the best convergence with respect to the basis size. Our calculations indicate that radial function bases are very promising in the context of multivariable integral equations.     

Dimer statistics on the Möbius strip and the Klein bottle

Closed-form expressions are obtained for the generating function of close-packed dimers on a 2M×2N simple quartic lattice embedded on a Möbius strip and a Klein bottle. Finite-size corrections are also analyzed and compared with those under cylindrical and free boundary conditions. Particularly, it is found that, for large lattices of the same size and with a square symmetry, the number of dimer configurations on a Möbius strip is 70.2% of that on a cylinder. We also establish two identities relating dimer generating functions for Möbius strips and cylinders.     

Asymptotic properties of the Hitchin–Witten connection

Letters in Mathematical Physics - We explore extensions to $${{\,\mathrm{SL}\,}}(n,{\mathbb {C}})$$ -Chern–Simons theory of some results obtained for $${{\,\mathrm{SU}\,}}(n)$$...     

Coupled-Channels Faddeev Equations for the K ??+?d Break-Up Reaction and the Shape of the ��(1405) Resonance

Coupled-channels Faddeev equations are derived for the K ^? + d break-up reaction with inclusion of the isospin breaking effect of the ${\bar{K}^0-K^-}$ mass difference.     

X-ray intensities following the fusions of dμd,dμt and pμd

Muonic X-ray spectra of μ³He and μ⁴He from dμd, pμd, and dμt fusion are analyzed for various target densities. Theoretical values of the intensity ratio μ³He(3→1)/μ³He(2→1) for dμd and pμd fusion at liquid hydrogen density are, respectively, 0.132 and 0.0553. These are in good agreement with the experimental results.     

The Hamiltonian Structure of the Nonlinear Schrödinger Equation and the Asymptotic Stability of its Ground States

In this paper we prove that ground states of the NLS which satisfy the sufficient conditions for orbital stability of M. Weinstein, are also asymptotically stable, for seemingly generic equations. The key issue is to prove that a certain coefficient is non-negative because is a square power. We assume that the NLS has a smooth short range nonlinearity. We assume also the presence of a very short range and smooth linear potential, to avoid translation invariance. The basic idea is to perform a Birkhoff normal form argument on the hamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0 solution for NLKG. But in our case, the natural coordinates arising from the linearization are not canonical. So we need also to apply the Darboux Theorem. With some care though, in order not to destroy some nice features of the initial hamiltonian.     

Experimental Results on Ultrahigh-Energy Cosmic Rays and Asymptotic Behavior of the Total Cross Section for Nucleon–Nucleon Interaction

Results obtained by studying the energy dependence of the total cross section for nucleon–nucleon interaction are presented. The analytic parametrization proposed within axiomatic quantum field theory describes quantitatively a unified set of experimental data on proton–proton and antiproton–proton scattering. At collision energies above 86 GeV, the parameter values fitted to subsets of all available data and accelerator data alone were found to deviate from the respective asymptotic values. This indicates that the Froissart–Martin limit for the total cross section has not yet been reached in the case of these subsets. An approximation of the total nucleon–nucleon cross section measured in cosmic rays leads for some parameters to fitted values that, within 1.0 to 1.3 standard deviations, agree with the respective asymptotic values. This resultmay be viewed as an indication of the beginning of the asymptotic functional behavior of the total nucleon–nucleon cross section measured in cosmic rays of ultrahigh energy O(100 TeV). This is confirmed within the color glass condensate approach.     

Existence and Asymptotic Stability of Periodic Solutions of the Reaction–Diffusion Equations in the Case of a Rapid Reaction

A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.     

Entanglement witnesses for d ⊗ d systems and new classes of entangled qudit states

We provide a new class of entanglement witnesses for d ⊗ d systems (two qudits). Our construction generalizes the one proposed recently by Jafarizadeh et al. for d = 3 and d = 4 on the basis of semidefinite linear programming. Moreover, we provide a new class of PPT entangled states detected by our witnesses which generalizes well known family of states constructed by Horodecki et al. for d = 3.     

Asymptotic Quasinormal Modes of the Garfinkle-Horowitz-Strominger Dilaton Black Hole

Using the monodromy technique proposed by Motl and Neitzke (.Adv. Theor. Math. Phys. 7(2003)307), we investigate the analytic forms of the asymptotic quasinormal frequencies for the massless scalar perturbation in the Garfinkle-Horowitz-Strominger dilaton spacetime. We find that the real parts of the quasinormal frequencies are TH ln 3. This agrees with that of the quasinormal modes in the Schwarzschild spacetime. Our result implies that Hod‘s conjecture about In 3 is still valid for the black hole spacetime in the string theory.