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The modified Newton-Sabatier method is extended to the inverse scattering problem for coupled channels at fixed energy. The coupled Schrödinger equations are assumed to depend on a potential matrix which couples the channels. This potential matrix is taken as independent on the angular coordinates of the relative motion. In order to test the inversion method it is applied to analytic S-matrices corresponding to potential matrices consisting of square well potentials. The scattered particles can be neutral or charged. In the latter case the S-matrix with the asymptotic Coulomb potential is transformed to the S-matrix with an asymptotic constant potential and then the inverse scattering problem for neutral particles is solved. [ABSTRACT FROM AUTHOR]

Applying the techniques of supersymmetric quantum mechanics we determine closed algebraic expressions for potentials that are phase-equivalent with the generalized Pöschl-Teller potential. Among the examples we discuss the elimination of any single bound state, adding a single bound state at specific energies and eliminating the first few bound states. In our work we applied the abstract mathematical formalism developed recently for the modification of the spectrum of potentials without changing the phase shifts, and adapted it to the case of the generalized Pöschl-Teller potential. We discuss the importance of shape invariance in these procedures and comment on the possibility of deriving similar closed formulas for various other potentials. [ABSTRACT FROM AUTHOR]

Based on a realization found previously for the Lie-algebra of the group SO(3, 1) a new realization for the SO(3, 2) algebra is presented. The symmetry generators are matrix-valued differential operators. The matrix-valued nature of the realization is induced by a finite dimensional (non-unitary) representation of the noncompact SO(3, 1) subgroup. The Casimir operators can be calculated, and the quadratic Casimir operator is shown to yield a three dimensional scattering problem with Pöschl-Teller potentials with an LS term, and an optical potential. The presence of the optical potential can be traced back to the noncompact nature of the SO(3, 1) group giving us the inducing representation. It is also pointed out, that the quadratic Casimir can be written in the instructive form of a Laplace-operator incorporating so(3, 1)-valued gauge-fields. The role played by gauge transformations in deriving new interaction terms is emphasized. [ABSTRACT FROM AUTHOR]

The generalized technique of Bargmann potentials is applied for the reconstruction of time-dependent and time-independent two-dimensional potentials and corresponding solutions in a closed analytic form on the basis of the inverse scattering problem in the adiabatic representation. Matrix elements of the induced gauge potentials can be constructed and studied in terms of obtained exact solutions. The approach suggested permits investigation of the dynamical quantum transition amplitudes for spectral data with a prescribed dependence on parametric coordinate variables. [ABSTRACT FROM AUTHOR]

A wide class of two-dimensional exactly solvable models is constructed on the basis of the inverse scattering problem in the adiabatic representation. Exactly solvable models with prescribed spectral properties are constructed by using the generalized method of Bargmann potentials. Two-dimensional potentials are presented within the consistent formulation of both mutually connected inverse problems to which the initial task is reduced: the parametric problem and the multichannel problem for the system of gauge equations. The algebraic technique is elaborated for the reconstruction of time-dependent and time-independent two-dimensional Bargmann potentials and corresponding solutions in a closed analytic form on the basis of the nonstandard parametric inverse problem with scattering data depending on a coordinate variable. Specific examples of exactly solvable models are given within the parametric problem on the entire line and on the half-line. In particular, transparent symmetric and nonsymmetric potentials, parametric family of phase-equivalent potentials, two-dimensional potentials without and with bound states are presented with the corresponding solutions of the parametric problem. [ABSTRACT FROM AUTHOR]

Potentials providing the same complex phase shifts as a given complex potential but with a shallower real part are constructed with supersymmetric transformations. Successive pairs of transformations eliminate normalizable solutions corresponding to complex eigenvalues of the Schrödinger equation with the full complex potential. With respect to real potentials, a new feature is the occurrence of normalizable solutions with complex energies presenting a positive real part. Removing such solutions provides a way of suppressing narrow resonances but may lead to complicated equivalent potentials with little physical interest. We discuss the singularity of the transformed potential and its relation with the Levinson theorem, the transformation of the Jost function, and the link with the Marchenko approach. The technique is tested with the solvable Pöschl-Teller potential. As physical applications, deep optical potentials for the α + 16O and 16O + 16O scatterings are transformed into l-dependent phase-equivalent shallow optical potentials. [ABSTRACT FROM AUTHOR]

A geometrical mapping of the semimicroscopic algebraic cluster model (SACM) is given. The geometrical variables are the relative radius vector, the quadrupole deformation parameters β and γ and the orientation of the deformed nucleus in the molecular system. The position of the minimum of the nuclear molecular potential is determined by the minimal number of π-bosons, describing the relative motion. This minimal number is determined by the implementation of the Pauli-principle. Applications to simple systems (16O + α, 12C + α and 28Si + 28Si) are presented. [ABSTRACT FROM AUTHOR]

The algebraic coupled-channels formalism based on SO(3,1) symmetry, originally proposed by Alhassid and Iachello (1989) for the description of heavyion reactions in the Fraunhofer diffraction regime, appears to be applicable also to reactions in the Fresnel regime, at the cost of some modifications, whose importance increases with decreasing energy and is crucial in the proximity of the Coulomb barrier. [ABSTRACT FROM AUTHOR]

The similarities and differences between the Algebraic Scattering Theory and the (Semimicroscopic) Algebraic Cluster Model are discussed. Based on this comparison suggestions are made for new kind of applications and for an alternative formulation of the scattering model. [ABSTRACT FROM AUTHOR]

The role of spectrum generating algebras and dynamic symmetries in the study of scattering processes is briefly reviewed. [ABSTRACT FROM AUTHOR]

The hierarchy of the periodic Toda lattices is derived by means of the Appelle transformation. Members of the hierarchy are determined by equations of motion, by Hamiltonians and Lax pairs. Only two- and three-particle systems besides the basic Toda lattice are solved by quadratures in accordance with the Stäckel theorem. Other members of the hierarchy assume the solving of the polynomial equation, the degree of which is higher than four. [ABSTRACT FROM AUTHOR]

We present an improvement of the Kato-Putnam theorem. We apply this to the spectral theory of some classes of generalized differential operators. [ABSTRACT FROM AUTHOR]

We demonstrate that the application of the supersymmetric (SUSY) transformation to a periodic potential in one dimension, which creates a bound state below the lowest allowed band, results in the unexpected phenomenon of potential reversal. This does neither alter other characteristics of the spectrum nor does it result in a reflection of Bloch waves. [ABSTRACT FROM AUTHOR]

Using recently developed inversion techniques based on generalized Darboux transformations we constructed, in uncoupled channels, momentum dependent potentials having a momentum independent term and one term linear in the square of the momentum. Varying the momentum dependent term of these potentials we created via inversion sets of potentials that are phase equivalent up to at least 2 GeV to the commonly used Nijmegen and Paris nucleon-nucleon potentials. These sets of potentials offer us the possibility to study the sensitivity of the triton binding energy on the momentum dependence of the NN-force. [ABSTRACT FROM AUTHOR]

Phase shifts derived from experimental cross section data on the elastic scattering of π+ and π− on low mass, zero spin and zero isospin nuclei are used to obtain model independent pion-nucleus potentials with a modified fixed energy inverse scattering formalism. No systematic difference between the π+ and π− potentials suggests the absence of any charge symmetry violating effect in the pion-nucleus interaction. [ABSTRACT FROM AUTHOR]

The modified Newton-Sabatier method is applied to invert electronatom scattering phase-shifts into effective potentials. The phase-shifts are corrected for the dipole polarisation interaction of the form −α/2r4. Polarisation phase shifts are calculated by the method of Holzwarth by using Mathieu functions of the second kind. The inversion potentials are compared with the potentials determined earlier by inverting the corresponding total phase-shifts. Examples involving synthetic as well as experimental phase-shifts show that the new method is capable of determining electron-atom potentials from a substantially smaller set of data at the cost of a prior determination of the underlying complex polarisation phase-shifts. [ABSTRACT FROM AUTHOR]

Two-body interactions of elementary particles are useful in particle and nuclear physics to describe qualitatively and quantitatively few- and many-body systems. We are extending for this purpose the quantum inversion approach for systems consisting of nucleons and mesons. From the wide range of experimentally studied two-body systems we concentrate here on πN, ππ, K+N, Kπ and $$K\bar K$$ . As input we require results of phase shift analyses. Quantum inversion Gelfand-Levitan and Marchenko single and coupled channel algorithms are used for Schrödinger type wave equations in partial wave decomposition. The motivation of this study comes from our two approaches: to generate and investigate potentials directly from data by means of inversion and alternatively use linear and nonlinear boson exchange models. The interesting results of inversion are coordinate space informations about radial ranges, strengths, long distance behaviors, resonance characteristics, threshold effects, scattering lengths and bound state properties. [ABSTRACT FROM AUTHOR]

The notion of interacting elementary particles for low and medium energy nuclear physics is associated with definitions of potential operators which, inserted into a Lippmann-Schwinger equation, yield the scattering phase shifts and observables. In principle, this potential carries the rich substructure consisting of quarks and gluons and thus may be deduced from some microscopic model. In this spirit we propose a boson exchange potential from a nonlinear quantum field theory. Essentially, the meson propagators and form factors of conventional models are replaced by amplitudes derived from the dynamics of self-interacting mesons in terms of solitary fields. Contrary to deduction, we position the inversion approach. Using Gel'fand-Levitan and Marchenko inversion we compute local, energy-independent potentials from experimental phase shifts for various partial waves. Both potential models give excellent results for on-shell NN scattering data. In the off-shell domain we study both potential models in (p, pγ) bremsstrahlung, elastic nucleon-nucleus scattering and triton binding energy calculations. It remains surprising that for all observables the inversion and microscopic meson exchange potentials are equivalent in their reproduction of data. Finally, we look for another realm of elementary interactions where inversion and meson exchange models can be applied with the hope to find more sensitivity to discern substructure dynamics. [ABSTRACT FROM AUTHOR]

We present a new numerical method for solving the problem of the reconstruction of interaction potential by a phase shift given on a set of closed intervals in (l,k)-plane, satisfying certain geometrical "Staircase Condition". The method is based on the Variable Phase Approach and on the modification of the Continuous Analogy of the Newton Method. [ABSTRACT FROM AUTHOR]

At the example of 12C-12C elastic scattering it is shown that the S-matrix at non-integer values of the orbital angular momentum quantum number contains valuable information which should be taken into account in the analysis of scattering data at fixed energy. The possibility of the available global inversion procedures to include such non-standard S-matrix values is discussed. Specifically, an extended Newton-Sabatier method based on the interpolation formulae of Sabatier is worked out and subsequently used to demonstrate the role of this information to reduce the inherent ambiguities of the inverse scattering problem at fixed energy. [ABSTRACT FROM AUTHOR]

The application of the one-dimensional inverse scattering problem to neutron and x-ray specular reflection is discussed. Consideration is given to the problems of the numerical treatment as well as to the limitations on the input, in particular, the phase problem. [ABSTRACT FROM AUTHOR]

The one-dimensional coupled-channel Marchenko equation in the presence of thresholds is derived. Various aspects of this equation are discussed and a numerical algorithm for its solution is proposed. The efficiency of the algorithm is demonstrated using simulated scattering data. [ABSTRACT FROM AUTHOR]

This report is a review of the qualitative theory of quantum design: about our discoveries of new features of quantum systems by visualizing all possible potential and wave function transformations for variations of a complete set of independent spectral and scattering parameters. Special attention is paid to recently revealed multichannel aspects of this theory. [ABSTRACT FROM AUTHOR]

In this paper we consider several separate questions for one-dimensional-Schrödinger operators on the entire line. The paper consists of an Introduction, seven short Sections and three Appendices. Each of these Sections and Appendices is almost independent. [ABSTRACT FROM AUTHOR]