Asymptotics of the S-matrix and unitarisation

The manner in which the elastic scattering amplitude obeys unitarity, how it enters the circle of unitarity, and what its asymptotic limit is, remains a problem for models which include terms that rise fast with s. We have checked that the features of cross sections which come from unitarisation are present for most unitarisation schemes, e.g. those that saturate the profile function or those that describe multiple exchanges via an analytic formula. We have also obtained a scheme which interpolates between different classes of the unitarisation and found corresponding non-linear equations. Considering different forms of energy dependence of the scattering amplitude, and a variety of unitarisation schemes, we show that, in order to reproduce the data, the fits choose an amplitude that corresponds to an asymptotic value S = 0.     

Asymptotics of resonances for 1D Stark operators

We consider the Stark operator perturbed by a compactly supported potential on the real line. We determine the forbidden domain for resonances, asymptotics of resonances at high energy and asymptotics of the resonance counting function for large radius.     

A priori estimates for the Hill and Dirac operators

The Hill operator Ty = −y″ + q′(t)y is considered in L ²(ℝ), where q ∈ L ²(0, 1) is a periodic real potential. The spectrum of T is absolutely continuous and consists of bands separated by gaps. We obtain a priori estimates of gap lengths, effective masses, and action variables for the KDV equation. In the proof of these results, the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator is used. Similar estimates for the Dirac operator are obtained.     

L 2-index formulae for perturbed Dirac operators

The Callias index theorem is generalized from the Euclidean case to certain spin manifolds with warped ends, making use of certain index-preserving deformations.     

The classical-limit S-matrix for Coulomb excitation

An expression for the classical-limit S-matrix for Coulomb excitation is derived and directly evaluated without resorting to stationary phase integration methods. The results obtained are in quantitative agreement with quantum mechanical calculations. This agreement and the simplicity of the method suggest tha feasibility of extending it to cases where other methods are not easily applied.     

The S-matrix for abstract scattering systems

Let S() be the S-matrix at energy for an abstract scattering system. We derive a bound, in terms of the interaction, on integrals of the form h () S()- _HS ² d, where denotes the Hilbert-Schmidt norm.Supported by the Swiss National Science Foundation.     

Semiclassical S-matrix theory for atomic fragmentation

A semiclassical scattering approach is developed which can handle long-range (Coulomb) forces without the knowledge of the asymptotic wave function for multiple charged fragments in the continuum. The classical cross section for potential and inelastic scattering including fragmentation (ionization) is derived from first principles in a form which allows for a simple extension to semiclassical scattering amplitudes as a sum over classical orbits and their associated actions. The object of primary importance is the classical deflection function which can show regular and chaotic behavior. Applications to electron impact ionization of hydrogen and electron–atom scattering in general are discussed in a reduced phase space, motivated by partial fixed points of the respective scattering systems. Special emphasis, also in connection with chaotic scattering, is put on threshold ionization. Finally, motivated by the reflection principle for molecules, a semiclassical hybrid approach is introduced for photoabsorption cross sections of atoms where the time-dependent propagator is approximated semiclassically in a short-time limit with the Baker–Hausdorff formula. Applications to one- and two-electron atoms are followed by a presentation of double photoionization of helium, treated in combination with the semiclassical S-matrix for scattering.     

S-matrix derivation of the Weinberg Model

By applying only the requirement of asymptotic softness (i.e., the requirement that amplitudes should not grow, asymptotically, faster than that in a renormalizable theory) to various Born amplitudes with four external lines and making some specific assumptions about the number of added particles to meet the requirement, we obtain all of the interaction Lagrangian and relations among the masses and coupling constants of the Weinberg Model.     

On the gauge independence of the S-matrix

The S-matrix is invariant with respect to the variation of any (global) parameter involved in the gauge-fixing conditions, if that variation is accompanied by a certain redefinition of the basis of polarization vectors. Renormalizability of the underlying gauge theory is not required. The proof is nonperturbative and, using the “extended” BRS transformation, quite simple. Received: 27 April 2001 / Published online: 6 July 2001     

Total S-matrix and adiabatic amplitudes for the three-body problem

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.     

The Dirac S-matrix in the k-plane

Expressions of the S-matrix, for the Dirac case, which are suitable for analytic continuation over the complex k-plane for a large class of potentials, fulfilling conditions (1), are derived.     

Asymptotic planarity: An S-matrix basis for the Okubo-Zweig-Iizuka rule

A mechanism is exhibited that monotonically depresses the cylinder component of the topological expansion with increasing t, and it is conjectured that all non-planar S-matrix components diminish as t increases, exchange degeneracy and the Okubo-Zweig-Iizuka rule becoming more accurately satisfied. Such asymptotic planarity is compared to the field-theoretical concept of asymptotic freedom. The characteristics low-t cylinder “quenching interval” is found to be the inverse of the mean value over a two-reggeon loop, of $\text{1}\text{2}\text{π}{}^2\text{(α′)}{}^2\text{(t}{}_1\text{? t}{}_2\text{)}{}^2\text{/(?t),}\text{where}\text{t}{}_1\text{and}\text{t}{}_2$ are the squared masses of the loop reggeons and α′ is the trajectory slope. For leading trajectories the low-t cylinder quenching interval is predicted by this formula to be roughly 0.5 GeV²-consistent with the observed pomeron intercept and slope, with the p-ω and f-A₂ mass differences and with the (φ,ω) deviation from ideal mixing. As t grows negatively over a corresponding interval, it is predicted that the pomeron will become nearly a pure SU(3) singlet. If the pion mass helps to set the scale for reggeon loops coupled to unnatural-parity trajectories, the cylinder quenching interval will be larger, explaining the large (η, η′) deviation from ideal mixing as well as the large π-η mass difference. Even when the small-t cylinder quenching is rapid (“precocious planarity”) the large-t approach to the planar limit turns out to be gentle. A by-product of this study is an explanation of the approximate reality and linearity of trajectories at large t.     

On Asymptotics for the Airy Process

The Airy process t→A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s ₁,s ₂, and t for the probability Pr(A(0)≤s ₁, A(t)≤s ₂). Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion for this probability as t→∞, with fixed s ₁ and s ₂. We shall show that the expansion can be obtained by using the Fredholm determinant representation for the probability. The main ingredients are formulas obtained by the author and C. A. Tracy in the derivation of the Painlevé II representation for the distribution function F ₂ plus a few others obtained in the same way.     

S-matrix theory and the density matrix formalism

A general equation governing the time development of the diagonal part of the density matrix is proposed for weakly interacting systems possessing no off-diagonal long range order. This equation, which involves generalized Møller operators of the type employed in S-matrix theory, is solved for two cases and leads to the generalized Pauli and Boltzmann equations.     

Diagonal structure of the S-matrix in the lee model

The diagonal structure of the S-matrix in the Lee model is studied. The N2θ- and N3θ-type eigenvalues are shown to factor exactly into the products of the corresponding Nθ eigenvalues, and it follows that these eigenvalues are free of production thresholds. The (implicit) eigenvalue equation for the V2θ-type eigenvalue is given, and this eigenvalue is shown to factor asymptotically into the product of the corresponding Vθ eigenvalues. These properties are conjectured to hold for Njθ- and Vjθ-type eigenvalues. It is conjectured that the two-body type eigenphase operator gives the dominant contribution to the S-matrix at high energy.     

Integrability of the S-matrix versus integrability of the Hamiltonian

This report reviews the relations between the integrability properties of the S-matrix and of the Hamiltonian. Particular emphasis is put on the situation where the Hamiltonian has a conserved quantity which is not compatible with the asymptotics and where correspondingly the integrability does not transfer to the S-matrix. As questions of integrability are more readily handled in classical dynamics, all developments are first performed classically. Several examples are discussed to illustrate the main points. The quantum mechanical discussion reveals that the eigenphase statistics of the S-matrix depends principally on the chaoticity of the scattering map while basis dependent quantities such as the distribution of matrix elements tend to have random matrix behaviour only in the presence of topological chaos. The relevance of these considerations to the evaluation of scattering data is discussed.     

Asymptotics for Bosonic atoms

It was proved by Benguria and Lieb that for an atom where the electrons do not satisfy the exclusion principle, the critical electron number N _c, i.e., the maximal number of electrons the atom can bind, satisfies lim inf_zN_c/Z 1 + , where Z is the nuclear charge. Here is a positive constant derived from the Hartree model. We complete this result by proving that the correct asymptotics for N _c(Z) is indeed _zN_c/Z = 1 + .This work was done while the author was a graduate student at Princeton University supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03.