On trigonometric intertwining vectors and non-dynamical R-matrix for the Ruijsenaars model

We elaborate on the trigonometric version of intertwining vectors and factorized L-operators. The starting point is the corresponding elliptic construction with Belavin's R-matrix. The naive trigonometric limit is singular and a careful analysis is needed. It is shown that the construction admits several different trigonometric degenerations. As a by-product, a quantum Lax operator for the trigonometric Ruijsenaars model intertwined by a non-dynamical R-matrix is obtained. The latter differs from the standard trigonometric R-matrix of A_n type. A connection with the dynamical R-matrix approach is discussed.     

A new treatment of correlations in reaction theory

The problem of correlations in reaction theory is treated making use of relationships between theS-matrix and appropriate correlation functions. The correlation functions are calculated from the resolvent matrix with the help of Feshbach's projection operator technique, generalized in such a way that correlations can be incorporated. A perturbation procedure in terms of a continued fraction expansion can be defined which avoids the well known divergence problems of the Born series. As a byproduct a conceptually and numerically simple treatment of single particle resonances is obtained and worked out in an example.     

Application of infinite oscillator shell model calculations to the description of nuclear reactions

A simple R-matrix procedure is proposed to provide a means of using nuclear structure information, from calculations in an infinite oscillator well, to describe reaction cross sections. The procedure uses values for the channel radius parameters of the R-matrix theory that are extracted from knowledge of the structure and reaction properties of the neighbouring closed shell nucleus answering one of the main criticisms expressed concerning the dependence of the R-matrix on unphysical parameters. Application of the model to the scattering of neutrons from ¹⁵N and comparison to previous theoretical predictions are considered.     

Irreducible multiplier corepresentations and generalized inducing

Wigner's classification of irreducible corepresentations into three types is generalised to irreducible multiplier corepresentations. Representations of Types I, II, and III have commutants isomorphic toR,H, andC, respectively. The more general problem of relating irreducible multiplier corepresentations of a group to those of an invariant subgroup is considered, and some algebraic aspects of “generalized inducing” are described. The Wigner classification is then re-obtained as a very simple instance of the general theory.     

The scattering of strings by a black hole

We study the elastic and inelastic scattering of strings by a Schwarzschild's black hole. Pair creation takes place (and in all the modes) as a consequence of the composite structure (oscillator modes) of the string. The S-matrix amplitudes including the pair-creation rate are found at first order in √α'/R_S (√α' = Planck's length and R_S = Schwarzschild's radius). Explicit computations are made in the weak-field expansion in powers of (R_S/b)^D−3 (b = impact parameter of the string center of mass). The deflection angle and cross sections are found. The quantum string corrections to the gravitational analogue of Rutherford's scattering are computed (these are of order α'²). The pair-creation or radiation amplitude we find here is of order α', it is non-thermal and of different origin than Hawking radiation.     

Alpha-Zerfall als Resonanz-Phänomen

Asuming exponential decay a new expression for the decay width is derived. A method to obtain the “prepared state” is given. The reduction of the many body to the one body problem is described. The resulting theory is compared with theR-matrix theory. It is applied onα-decay in¹⁶O.     

Photodisintegration of 13C

The nucleon decay of the photoresonance in ¹³C is analyzed using bound-state shell model and R-matrix theory. Total and partial photo cross sections, branching ratios, final state population, and nucleon emission spectra are considered and compared with available data.     

The 17O(p, α)14N reaction: Physics and astrophysics

The nucleus ¹⁸F has been studied using the ¹⁷O(p,α)¹⁴N reaction over the energy range E_p = 400 to 1400 keV with thin Ti¹⁷O₂ targets. New resonances at E_p = 525, 688, 1015, 1276 and 1338 keV have been discovered. The strengths for all resonances in this energy range have been measured and the entire cross section in this region has been fitted using the R-matrix formalism of Lane and Thomas. Some discussion is presented concerning the values of the boundary condition parameters used in the R-matrix fit and two sets of results are presented corresponding to two different treatments of the boundary conditions.The ¹⁷O(p, α)¹⁴N cross section has been extrapolated to astrophysical energies using an R-matrix calculation incorporating the partial widths obtained from the data from E_p = 400 to 1400 keV. These results are compared with previous ¹⁷O(p, α)¹⁴N calculations and the ¹⁷O(p, γ)¹⁸F direct capture calculation of Rolfs and Rodney.     

Extraction of spectroscopic factors using R-matrix theory

The relation of the dimensionless reduced width of a state to its spectroscopic strength is discussed and the spectroscopic factor is calculated as the ratio of the reduced width to that of an optical potential analyzed with the same channel radius and boundary condition number. From a model square well problem, the reduced width is shown to be sensitive to these parameters. Neutron scattering to the 5.08 MeV state of ¹⁷O is analyzed with R-matrix theory and the value obtained for its spectroscopic factor is 0.80. This value is shown to be moderately independent of the parameters of the optical potential, a fact which greatly simplifies analysis by eliminating the strong dependence upon the location of the pure single particle state.     

Fine-structure splitting calculation in the Ar III ion: comparison of perturbative (Breit-Pauli) and non-perturbative (MCDF-EAL) predictions

We have calculated fine structure splitting in the Ar III ion using the R-matrix method. Two independent atomic structure calculations have been performed. Results from the Breit-Pauli ?C and the Dirac-Atomic ?C R-matrix relativistic calculations are analysed comparatively. Cross sections and collision strengths are provided for selected weak and intercombination transitions, allowing explicitly for resonance effects. Convergence of the partial wave expansion is ensured by examining the partial collision strengths at collision energies up to 20 Ry. Radiative data are also reported here. We discuss all these results and compare them, when possible. For the allowed transitions, for which the resonance structure is not dominant, the two calculations are nearly identical.     

Hopf-type deformed oscillators,their quantum double and a q-deformed Calogero-Vasiliev algebra

A q-deformed oscillator Hopf algebra is presented and the quantum double construction is carried out to obtain an R-matrix. Investigation of the algebra's structure and Fock-type representation leads to a new q-deformed Calogero-Vasiliev algebra.     

Dual models as saddle point approximations to Polyakov's quantized string

We show that if the metric becomes singular on the boundary, then Polyakov's quantized string theory has a saddle point. This leads to an off-shell Green function, the S-matrix of which is the standard dual (Veneziano) model.     

Reduction of the Lane-Robson “calculable” reaction theory to standard R-matrix form

General dynamical equations derived from the Lane-Robson calculable reaction formalism are cast into a form amenable to standard R-matrix treatment, permitting the resonance content of the equations to be made explicit. Formulae are given which enable the collision matrix and the amplitudes of physical eigenfunctions to be calculated directly from the R-matrix with or without the isolation of resonance contributions. The present methodology permits a significant reduction of effort in numerical investigations of the energy dependence inherent in dynamical models of the nucleus. The formalism is illustrated by calculational results obtained from a potential model fitted to ¹⁶O + n scattering data.     

Renormalizability and asymptotic freedom in quantum gravity

We discuss a previously proposed renormalizable theory of gravity involving R²_μν, and N massless fermion (vector boson) fields in which the unitarity problem is resolved within a $\text{1}\text{N}$ expansion. The infrared limit is precisely Einstein's theory, but the high-energy behavior is determined by the dimensionless, asymptotically free coupling of the R²_μν. Various attractive possible consequences of the theory are pointed out.     

Integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons

A general graded reflection equation algebra is proposed and the corresponding boundary quantum inverse scattering method is formulated. The formalism is applicable to all boundary lattice systems where an invertible R-matrix exists. As an application, the integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons are investigated. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary system is solved by means of the coordinate space Bethe ansatz technique and the Bethe ansatz equations are derived. As a sideline, it is shown that all R-matrices associated with a quantum affine superalgebra enjoy the crossing-unitarity property.     

Holomorphic curves from matrices

Membranes holomorphically embedded in flat non-compact space are constructed in terms of the degrees of freedom of an infinite collection of 0-branes. To each holomorphic curve we associate infinite-dimensional matrices which are static solutions to the matrix theory equations of motion, and which can be interpreted as the matrix theory representation of the holomorphically embedded membrane. The problem of finding such matrix representations can be phrased as a problem in geometric quantization, where ϵ ∝ l_P³/R plays the role of the Planck constant and parametrizes families of solutions. The concept of Bergman projection is used as a basic tool, and a local expansion for the action of the projection in inverse powers of curvature is derived. This expansion is then used to compute the required matrices perturbatively in ϵ. The first two terms in the expansion correspond to the standard geometric quantization result and to the result obtained using the metaplectic correction to geometric quantization.     

The dual (p,q)-Alexander-Conway Hopf algebras and the associated universal ℐ-matrix

The dually conjugate Hopf algebrasFun _p,q (R) andU _p,q (R) associated with the two-parametric (p,q)-Alexander-Conway solution (R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebraU _p,q (R) is extracted. The universal ?-matrix forsFun _p,q (R) is derived. While expressing an arbitrary group element of the quantum group characterized by the noncommuting parameters in a representation independent way, the ?-matrix generalizes the familiar exponential relation between a Lie group and its Lie algebra. The universal ?-matrix and the FRT matrix generators,L ^(±), forU _p,q (R) are derived from the ?-matrix.     

Equivalent rotations associated with the permutation inversion group revisited: symmetry projection of the rovibrational functions of methane

In this work the analysis of the equivalent rotations from the permutation inversion group formalism is revisited. We emphasize that explicit knowledge of changes in the Euler angles are not required in order to determine the transformation that a given symmetry operation causes to the rotational functions when dealing with the permutation inversion group formalism. Indeed, matrix elements of the equivalent rotations are provided by a single Wigner's D ^(j)(R) function. Taking advantage of this, we propose a symmetry projection approach to build the rovibrational functions of methane. This approach focuses on the relevance of the isomorphism between permutations and equivalent rotations. In our method, symmetry adapted functions are obtained by simultaneous diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner's D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits us to obtain in a systematic fashion an orthonormal set of symmetry-projected functions, with good total angular momentum, and carrying the irreducible representations of the molecular symmetry group.     

On the regular holonomic character of theS-matrix and microlocal analysis of unitarity-type integrals

The previously proved results that every analytically renormalized Feynman integral is a regular holonomic function suggests that theS-matrix should be locally expressible as an infinite sum of regular holonomic functions. A regularity propertyR is formulated that expresses the condition that theS-matrix be locally expressible near each physical pointp as a convergent sum of regular holonomic functions, with each term enjoying some of the regularity properties of a corresponding Feynman integral. This propertyR holds at every physical pointp that has yet been analyzed by the methods of axiomatic field theory orS-matrix theory. Some analyticity properties of unitarity-type integrals are then examined under the assumption that theS-matrix satisfies propertyR and a weak integrability condition. These results rest heavily on some recently proved properties of regular holonomic functions.     

Coupled-channel R-matrix method on a Lagrange mesh

The coupled-channel R-matrix method on a Lagrange mesh is a very simple approximation of the R-matrix method with a basis. The mesh points are zeros of shifted Legendre polynomials. Bound-state energies and scattering matrices are easily calculated with small numbers of potential values at mesh points. A test with an exactly solvable two-channel potential provides an excellent accuracy over a broad energy range with only 30 mesh points. The efficiency of the method is illustrated for a single channel on α + α scattering and for two channels on the deuteron ground-state energy and on nucleon-nucleon scattering.