Analyticity and dispersion relations in atomic scattering

We study the analyticity properties of forward electron-neutral atom scattering in a second quantized formalism. We derive a Low equation for these processes and deduce from it the location of direct and exchange bound state poles and the presence of both right- and left-hand cuts. We also examine the singularities produced by the exchange Born terms in a potential approximation and identify their physical origin. In a local theory, these Born term singularities are a result of the exchange of ionized atoms accompanied by a photon. Based on the singularity structure that we have obstained, we derive novel dispersion relations for forward electron-atom scattering. These equations contain additional terms not present in the “customary” form of dispersion relations for atomic scattering found in the literature. We comment on, and critically discuss, recent work done in this area. The possibility of anomalous thresholds is examined and they are seen to be irrelevant for these processes.     

Resonance eigenfunctions in chaotic scattering systems

We study the semiclassical structure of resonance eigenstates of open chaotic systems. We obtain semiclassical estimates for the weight of these states on different regions in phase space. These results imply that the long-lived right (left) eigenstates of the non-unitary propagator are concentrated in the semiclassical limit ħ → 0 on the backward (forward) trapped set of the classical dynamics. On this support the eigenstates display a self-similar behaviour which depends on the limiting decay rate.     

Analyticity properties of the Feigenbaum function

Analyticity properties of the Feigenbaum function [a solution ofg(x)=–^–1 g(g(x)) withg(0)=1,g(0)=0,g(0)<0] are investigated by studying its inverse function which turns out to be Herglotz or anti-Herglotz on all its sheets. It is found thatg is analytic and uniform in a domain with a natural boundary.     

Analyticity and clustering properties of unbounded spin systems

Transforming any lattice system in a polymer model, we use known analytic and cluster properties of the latter to derive similar ones for general lattice models with two-body interactions. These properties of the lattice model hold when the temperature is high enough.Supported by the Fonds National Suisse de la Recherche Scientifique     

Analytic properties of the scattering matrix for two-dimensionally periodic objects

The analytic properties of the scattering matrix for a two-dimensionally periodic object are investigated with specific reference to low-energy electron diffraction. The plane-wave basis is adopted and the scattering matrix is considered as a function of the normal component K of the propagation vector of an incident wave. The unitarity property of the scattering matrix, which guarantees conservation of flux in the absence of inelastic scattering, is extended in two ways: (1) inclusion of a relation governing evanescent as well as propagating waves, and (2) provision for an absorbing medium to represent the effect of inelastic scattering. The effect of absorption is represented by going to complex values of K.The main result is that the scattering matrix S satisfies a relation of the form

$\text{S(K}{}^1\text{)}{}^{\mathrm{?1}}\text{= [}\text{F}\text{+ GS(K)}{}^1\text{][G + FS(K)}{}^1\text{]}{}^{\mathrm{?1}}$

where K¹ denotes the complex conjugate of K on a specific unphysical sheet and F and G are matrices whose function is to project the propagating and evanescent parts of the wave field. A similar relation between S(?K) and S(K) is given.A numerical application to scattering of low-energy electrons by an atom layer is described.     

Analyticity properties of the weakly coupled double-well model

In this note we study lattice Φ⁴-models with Hamiltonian $$H = \tfrac{1}{2}(\varphi , - \Delta \varphi ) + \lambda \Sigma \left( {\varphi _i^2 - \frac{{m^2 }}{{8\lambda }}} \right)^2$$ and Gaussian boundary conditions. Using the polymer expansion we obtain analyticity of the pressure and the correlation functions in the infinite volume limit in a region $$\left\{ {\left. \lambda \right| \left| \lambda \right|< \varepsilon ,\left| {arg } \right.\left. \lambda \right|< \frac{\pi }{2} - \delta } \right\}$$ for every δ>0.     

Analyticity properties and many-particle structure in general quantum field theory

The extraction of one-particle singularities from then-point functions is performed in the framework of L.S.Z. field theory in the case of a single massive scalar field. It is proved that the “one-particle irreducible” functions thus obtained enjoy the analytic and algebraic primitive structure of generaln-point functions (up to a finite number of generalized C.D.D. singularities). Finally under an additional technical assumption, it is shown that the Glaser-Lehmann-Zimmermann relations stating the completeness of asymptotic states yield similar relations satisfied in any given channel by the corresponding one-particle irreducible functions.     

Analyticity properties and power law estimates of functions in percolation theory

We consider percolation on the sites of a graphG, e.g., a regulard-dimensional lattice. All sites ofG are occupied (vacant) with probabilityp (respectively,q=1–p), independently of each other.W denotes the cluster of occupied sites containing a fixed site (which will usually be taken to be the origin) andW the cardinality ofW. The percolation probability is the probability that #W=, i.e.,(p)=P _p{# W=}. Some critical values ofp,p _H andp _T, are defined, respectively, as the smallest value ofp for which(p)> 0, and for which the expectation of #W is infinite. Formally,p _H=inf {p(p)>0} andp _T=inf{p E _p{#W}=}. We show for fairly general graphsGthat ifp _T, thenP _P{#W n} decreases exponentially inn. For the special casesG =G ₀= the simple quadratic lattice andG ₁= the graph which corresponds to bond-percolation on ², we obtain upper and lower bounds for(p) of the formC¦p¦-P _H¦, and bounds forEp{#W} of the formC¦p–p _H¦^–. We also investigate smoothness properties of (p)=E _p{number of clusters per site} =E _p {(#W)^–1; (#W) 1}. This function was introduced by Sykes and Essam, who assumed that (·) has exactly one singularity, namely, atp=p _H. For the graphsG ₀ andG ₁, (i.e., site or bond percolation on ²) we show that (p) is analytic atp p _H and has two continuous derivatives atp=p _H. The emphasis is on rigorous proofs.Research supported by the NSF through a grant to Cornell University.     

Analyticity properties and many-particle structure in general quantum field theory

In the framework of L.S.Z. field theory in the case of a single massive scalar field, the two-particle irreducible parts of then-point functions (in any single channel and for arbitraryn) are defined as the solutions of a system of integral equations suggested by the perturbative framework. These solutions enjoy the analytic and algebraic properties of generaln-point functions (up to possible polar singularities of generalized C.D.D. type). Morever it is shown that the completeness of asymptotic states in the two-particle spectral region is equivalent to the analyticity of the two-particle irreduciblen-point functions in the corresponding regions of complex momentum space.     

Statistical properties of parameters of the scattering matrix for compound nuclear reactions

A numerical calculation for determining the statistical distributions of the partial level widths and S-matrix poles is performed. Many of the expected properties of these distributions are confirmed. It is found that the parameters of the optical background representation have more suitable statistical properties. A new relationship among the parameters of the optical background representation, with its numerical checks, is presented.     

Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity

The wave and scattering operators for the equation $$\left( {\square + m^2 } \right)\varphi + \lambda \varphi ^2 = 0$$ withm>0 and λ>0 on four-dimensional Minkowski space are analytic on the space of finite-energy Cauchy data, i.e.L ₂ ¹ (R ³)⊕L ₂(R ³).     

Analyticity properties of the surface free energy of the Ising model

For an Ising ferromagnet with nearest-neighbour interactions of strengthK and surface magnetic fieldh, the surface free energy in the presence of a positively (or negatively) magnetized zero-field bulk phase is shown to be analytic inh for Reh<K–/, where =2.96 ... and is the inverse temperature. This puts the lower boundK–/ on the values ofh at which wetting and layering transitions can take place.     

Analyticity properties of the Ising model in the antiferromagnetic phase

The partition function of the Ising antiferromagnet is proved to have no zeroes in an annulus around the origin in the complexz-plane. The intersection of this annulus with the positive real axis belongs to the antiferromagnetic region. The free energy and the correlation functions are analytic in the annulus.On leave of absence from the University of Groningen, the Netherlands; supported by the Netherlands Organization for Pure Scientific Research (Z.W.O.).Supported by the National Swiss Foundation for Scientific Research.     

Analyticity and renormalization group

We prove the inequalities ψ(y, α) ?α, |α_s(d/dα_s)(β(α_s)/α_s| ? 1 (for the Paterman-Stueckelberg-Gell-Mann-Low functions in QED and QCD) and γ₀(α_s ? 1 (for the anomalous dimension of the gauge-invariant operator O(x)). The consequences of the inequalities are discussed: for modern energies, comparison of theoretical and experimental moments of deep inelastic structure functions has a meaning only for N ? 7 (singlet case) and N ? 50 (non-singlet case); perturbation theory in QCD has a meaning only for α_rms ? 0.45.     

Analyticity and Lie generators

We consider and answer in the negative the question whether, given a Lie group representation, analyticity of a vector for the representatives, in the differentiated representation, of a set of Lie generators of the Lie algebra implies analyticity for the group representation.     

Analyticity properties of the correlation functions for the anisotropic Heisenberg model

It is shown for the Heisenberg model that the correlation functions are analytic inh andT if Re(h)≠0 andT is positive.     

Non-translation invariant Gibbs states with coexisting phases III: Analyticity properties

We consider the spatially inhomogeneous Gibbs states for the three dimensional Ising and Widom-Rowlinson models. We prove the analyticity inz=exp(–2J) for small |z| of the spin correlation functions of these Gibbs states and of the surface tension.Supported in part by N.S.F. Grant PHY 77-22302Supported by the Swiss National Foundation for Scientific Research     

Overlapping resonances and unitarity of the scattering matrix

An automatically unitary product expansion for a scattering matrix with arbitrary over-lapping resonances is discussed and used to derive sum rules for the resonance parameters. Problems with the implementation of time-reversal invariance are pointed out.     

Algebraic and geometric properties of Bethe Ansatz eigenfunctions on a pentagonal magnetic ring

The exact solution of the eigenproblem of the Heisenberg Hamiltonian for the XXX model in the case of a magnetic ring with N=5 nodes is presented in a closed algebraic form. It is demonstrated that the eigenproblem itself is expressible within the extension of the prime field Q of rationals by the primitive fifth root of unity, whereas the associated Bethe parameters, i.e. pseudomomenta, phases of scattering, and spectral parameters, require some finite field extensions, such that the nonlinearity remains algebraic rather than transcendental. Classification of exact Bethe Ansatz eigenstates in terms of rigged string configurations is presented.