Rigorous sum rules and bounds on pion-pion scattering amplitudes with positivity

We derive the complete set of crossing symmetric forward pion-pion amplitudes with non-negative absorptive parts obeying twice-subtracted dispersion relations. We give a method to eliminate subtraction constants for these amplitudes by making a simultaneous use of subtracted dispersion relations for the amplitude and its inverse, and unitarity. We thus deduce from axiomatic field theory (i) lower bounds on forward real parts in terms of physical forward amplitudes, (ii) lower bounds on S- and P-wave scattering lengths in terms of forward amplitudes in any part of the physical region, (iii) sum rules on scattering lengths in terms of physical forward amplitudes which provide a complete test of forward dispersion relations, (iv) upper bounds on scattering lengths in terms of total cross sections and (v) absolute sum rule inequalities on total and forward differential cross sections in terms of the pion mass alone which provide new tests of dispersion relations not involving scattering lengths.     

Rigorous bounds for the complex dielectric constant of a two-component composite

Rigorous bounds are systematically derived for the bulk effective dielectric constant of a two-phase composite material ε_e under various assumptions about the available information. The bounds are drawn for some special cases in the complex ε_e-plane. The question of optimality of the bounds and their connection to solvable microgeometries is discussed. It is shown how linear programming theory can be used to aid in the derivation of these bounds.     

Rigorous solution of a problem for a cylindrical cavity containing a ferrite

Conclusions Equation (6) relates the complex frequency for a TM_nmo mode in a cylindrical cavity to the geometrical parameters, ferrite parameters, and volume and position of the specimen.Formula (6) is a generalization of characteristic equations previously derived [3, 4], which are of great value.The dispersion equation of (6) is of interest in that it allows one to calculate the effects of the dielectric parameters on the cavity, which is of value if a large tuning range is needed.Numerical results are rather laborious to obtain, but some valuable conclusions can be drawn in a relatively simple fashion, as will be shown in a forthcoming paper.     

Rigorous solution of a problem for a cylindrical cavity containing a ferrite and a dielectric

A rigorous solution is given for modes of TM_nmo type. The complex resonant frequency _n is expressed via a transcendental characteristic equation. Some particular cases are discussed.Read at the Third All-Union Conference on Ferrites, Leningrad, 23 October 1963.     

Rigorous bounds for critical temperatures of O(N) spin models by transformations of block spin type

We obtain new upper bounds of critical temperatures of N-vector (Heisenberg) models. We apply a transformation of block spin type to random walk representations of the spin models, which was developed by Fröhlich et al. more than a decade ago. Though the transformation is applied just one time, the upper bounds are considerably improved.     

Deep inelastic scattering in improved lattice QCD (I). The first moment of structure functions

We present the complete 1-loop perturbative computation of the renormalization constants and mixing coefficients of the operators that measure the first moment of deep inelastic scattering structure functions, employing the nearest neighbor improved lattice QCD action. The interest of using this action in Monte Carlo simulations lies in the fact that all terms which in the continuum limit are effectively of order a (a being the lattice spacing) have been proven to be absent from [p]Because of the complexity of the calculations, we have checked the analytical expression of all Feynman diagrams using Schoonschip. To this end we have developed a suitable code designed to automatically carry out all the necessary lattice algebraic manipulations, starting from the elementary building blocks of each diagram. [p]We have found discrepancies with some of the published numbers, but we are in agreement with the known results on the energy-momentum tensor.     

Rigorous and approximate methods for modeling wave scattering from a locally perturbed perfectly conducting surface

Rigorous and approximate methods are considered for solving the problem of harmonic plane wave scattering from a plane surface arbitrarily perturbed along one dimension on a finite interval. This problem is treated using the Fredholm integral equations of the second kind and the Kirchhoff and Rayleigh approximations. The estimates of the computational efficiency of the integral equation method and the Rayleigh approximation are compared by calculating fields scattered from random rough surfaces in the resonance region (i.e., when the roughness height is comparable to or smaller than the incident wavelength) for an arbitrary incidence of a plane wave. Scattering patterns calculated using the integral equations and the Kirchhoff approximation are discussed in the case of large-scale random rough surface scattering. Particular attention is paid to scattering at near-grazing incidence.     

Construction of a unitary analytic scattering amplitude (I). Scalar particles

By imposing cos θ analyticity, we extend earlier investigations into the problem of constructing a unitary scattering amplitude from experimental measurements. The new approach is superior to earlier work, inasmuch as (a) the partial waves fall off exponentially, (b) we can handle cross sections with zeros, (c) we have a convergent Newton-Kantorovich interaction even when the fixed-point theorems do not apply, (d) we can study bifurcation points, (e) in the inelastic region, we find that analyticity does not remove the continuum ambiguity, (f) for quasi-elastic reactions (e.g. photoproduction) we resolve the Watson ambiguity for the phase shifts. We work at fixed energy, and throughout this first paper we neglect spin and isospin.     

Derivation and implications of rigorous absolute bounds for ππ partial waves up to 1 GeV

Generalized dispersion relations and axiomatic results are used to derive absolute upper bounds for the ππ partial waves in a complex energy domain containing the physical region up to 1 GeV. As applications, a new lower bound for the π⁰π⁰ S-wave scattering length is given and some ambiguities in the connection between resonances and second sheet poles are removed.     

On a method for solving the inverse problem in potential scattering

A method for solving the inverse problem in the non-relativistic elastic scattering theory, using the analytic and asymptotic properties of the scattering amplitude is proposed and the influence of the discontinuity parameters of the scattering amplitude on the properties of the resulting potentials is discussed. The case with spherically symmetric forces and without bound states is considered. The possibility for solving the inverse problem by this method, leading to the singular repulsive potentials is mentioned.     

The A(y) problem for p-3He elastic scattering

We present evidence that numerically accurate quantum calculations employing modern internucleon forces do not reproduce the proton analyzing power, A(y), for p- 3He elastic scattering at low energies. These calculations underpredict new measured analyzing powers by approximately 30% at E(c.m.) = 1.20 MeV and by 40% at E(c.m.) = 1.69 MeV, an effect analogous to a well-known problem in p-d and n-d scattering. The calculations are performed using the complex Kohn variational principle and the (correlated) hyperspherical harmonics technique with full treatment of the Coulomb force. The inclusion of the three-nucleon interaction does not improve the agreement with the experimental data.     

Microscopic R-matrix theory in a generator coordinate basis: (III). Multi-channel scattering

The microscopic R-matrix theory presented in two previous papers is extended to the multichannel scattering case. An antisymmetrized wave function is built in the two-centre harmonic oscillator model. This wave function is proved to be equivalent to the resonating group one even if the nuclei have non-zero spin. The method only requires the calculation of numerical values of matrix elements between Slater determinants. The microscopic R-matrix theory may be applied to study both reactions and inelastic scattering.     

The Riemann boundary problem on a torus and the inverse scattering problem for the Landau-Lifschitz equation

The Landau-Lifschitz equation for ferromagnets is written as a compatibility condition of two linear differential equations. The inverse scattering problem for this system is ruduced to the matrix Riemann boundary problem on a torus and then to a certain Fredholm integral equation. The N-soliton solitoons are also obtaine.     

Construction of a unitary analytic scattering amplitude: (IV). Many two-body channels

We extend previous results concerning the existence of unitary amplitudes that correspond to a given differential cross section to the many-channel case. We use cos θ analyticity and show that all of the one-channel results may be adapted to the more general problem. A new constraint, the commutavity of the dispersive and absorptive parts, is shown to be automatic in all but exceptional singular cases.     

The inverses-wave scattering problem for a class of potentials depending on energy

The inverse scattering problem is considered for the radials-wave Schrödinger equation with the energy-dependent potentialV ⁺(E,x)=U(x)+2 Q(x). (Note that this problem is closely related to the inverse problem for the radials-wave Klein-Gordon equation of zero mass with a static potential.) Some authors have already studied it by extending the method given by Gel'fand and Levitan in the caseQ=0. Here, a more direct approach generalizing the Marchenko method is used. First, the Jost solutionf ⁺(E,x) is shown to be generated by two functionsF ⁺(x) andA ⁺(x,t). After introducing the potentialV ^–(E,x)=U(x)–2 Q(x) and the corresponding functionsF ^–(x) andA ^–(x,t), fundamental integral equations are derived connectingF ⁺(x),F ^–(x),A ⁺(x,t) andA ^–(x,t) with two functionsz ⁺(x) andz ^–(x);z ⁺(x) andz ^–(x) are themselves easily connected with the binding energiesE _n ⁺ and the scattering matrixS ⁺(E),E>0 (the input data of the inverse problem). The inverse problem is then reduced to the solution of these fundamental integral equations. Some specific examples are given. Derivation of more elaborate results in the case of real potentials, and applications of this work to other inverse problems in physics will be the object of further studies.Physique Mathématique et Théorique, Equipe de recherche associée au C.N.R.S.     

Pion-deutron scattering in the Δ(1236) energy region as a three-body problem

We study pion-deuteron scattering in the first π-N resonance energy region, in a three-body model based on Faddeev's equations. We discuss the effects of multiple scattering, nuclear binding, and virtual excitation of the target, on the energy behavior of the cross section.     

Relativistic inverse scattering problem for a superposition of a nonlocal separable and a local quasipotential

Within the relativistic quasipotential approach to quantum field theory, the relativistic inverse scattering problem is solved for the case where the total quasipotential describing the interaction of two relativistic spinless particles having different masses is a superposition of a nonlocal separable and a local quasipotential. It is assumed that the local component of the total quasipotential is known and that there exist bound states in this local component. It is shown that the nonlocal separable component of the total interaction can be reconstructed provided that the local component, an increment of the phase shift, and the energies of bound states are known.