Relative Oscillation Theory,Weighted Zeros of the Wronskian,and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm–Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein’s spectral shift function is established. Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.     

Scattering Theory with Finite-Gap Backgrounds: Transformation Operators and Characteristic Properties of Scattering Data

We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give necessary and sufficient conditions for the scattering data in the case of perturbations with finite second (or higher) moment.     

On the electronic structure of liquid metals

We consider the electronic structure of liquid metals as a problem of multiple scattering. The total scattering operator is developed in terms of the reaction operators of a single scattering event, having a ‘standing wave’ boundary condition. This method has several advantages with respect to the approach where the total scattering operator is developed in terms of the scattering operators of a single scattering. Since the reaction operators are hermitian, the occurring perturbation expansions have a real perturbation parameter, which is more easily handled. Furthermore, if in the expansion of the total scattering operator, only terms up to the second order are maintained, this approximate expression satisfies already the ‘optical theorem’, just as the exact total scattering operator. Finally, spurious damping terms are eliminated at the outset. For very dense systems we obtain practically non-attenuated waves and this can be understood as if the damping due to a single scattering event is compensated by the reaction of the surrounding scatterers.     

Stationary Scattering Theory for a Charged Particles Transport Problem

We consider a nonautonomous transport problem, the modelization of the charge exchange dynamics in a monoatomic ionized gas, and apply scattering theory to its dynamics. The free dynamics corresponds to the evolution of the total distribution of particles (neutral plus ionized particles) and the perturbed dynamics corresponds to the evolution of the neutral particles, which is the solution of a nonautonomous transport problem. The existence of the time-dependent wave operators was proved by the first author. In the present paper we follow Howland's formalism in constructing a stationary scattering theory for this nonautonomous transport problem by studying the evolution equation. We prove the existence of the wave operators and by using the smooth perturbation technique we obtain the similarity between perturbed and unperturbed operators.     

Definition and existence of multichannel scattering states

The field theoretical formulation of quantum mechanics is used to consider the nonrelativistic multichannel scattering theory. With the help of appropriately constructed time dependent creation operators, Hilbert vectors are formed whose limits in time can be defined as multichannel scattering states in the usual sense. The existence of these states is proved under certain assumptions for the potential, by showing the convergence of the above mentioned operators. The commutation relations for the limits of these operators are given.     

Scattering theory relevant to the linear transport of particle swarms

The long-time behavior of the velocity distribution of a spatially uniform diluted guest population of charged particles moving within a host medium under the influence of a D.C. electric field is studied within the framework of scattering theory. We prove the existence of wave and scattering operators for a simplified one-dimensional model of the linearized Boltzmann equation. The theory is applied to the study of the long-term behavior of electrons and the occurrence of traveling waves in runaway processes.     

Polarized lepton-hadron scattering in asymptotically free gauge theories

We analyze polarized deep inelastic electron and neutrino scattering in asymptotically free gauge theories of the strong interactions. We compute the logarithmic deviations from Bjorken scaling in the moment sum rules for the structure functions due to the leading twist-2 and -3 operators that control the light-cone expansion. We also consider continuations to the elastic limit and to the Regge region.     

Algebraic solution of the Hubbard model on the infinite interval

We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite line at zero density. This enables us to diagonalize the Hamiltonian algebraically. The eigenstates can be classified as scattering states of particles, bound pairs of particles and bound states of pairs. We obtain the corresponding creation and annihilation operators and calculate the S-matrix. The Hamiltonian on the infinite line is invariant under the Yangian quantum group Y(su(2)). We show that the n-particle scattering states transform like n-fold tensor products of fundamental representations of Y(su(2) ) and that the bound states are Yangian singlet.     

Propagation of states in dilation analytic potentials and asymptotic completeness

We estimate the space-time behavior of scattering states for two-body Schrödinger operators with smooth, dilation analytic potentials. We use our estimates to give a simple proof of asymptotic completeness for a class of long-range potentials, including the Coulomb potential plus a fairly general short-range perturbation.Supported by USNSF Grant MCS-78-01885     

Classical scattering with long range forces

We discuss the classical two-body scattering problem for potentials which decrease at infinity liker ^?α, 1≧α>0. We prove existence and uniqueness theorems for scattering orbits parametrized by their asymptotic data. Wave operators are constructed and their properties discussed. We also discuss and prove cluster properties of theS-operator.     

QCD Interaction in Pion-Deuteron Elastic Scittering

The pion-deuteron elastic scattering matrix element is under consideration in the framework of QCD Theinteractions including theperturbative effect and nonperturbative contributions of gluon condensate and quark condensate in the scattering process are discussed. The interaction operators are given.     

Absolute summability of the trace relation for certain Schrödinger operators

A recently established general trace formula for one-dimensional Schrödinger operators is systematically studied in the context of short-range potentials, potentials which approach different spatial asymptotes sufficiently fast, and appropriate impurity (defect) interactions in one-dimensional solids. We prove the absolute summability of the trace formula and establish its connections with scattering quantities, such as reflection coefficients, in each case.This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material     

Influence of vacuum energy on scattering

We deal with photon-electron scattering which occurs between two uncharged conducting parallel plates moving away from each other at a constant velocity. The electromagnetic vacuum field between two plates is defined by the configuration of space and also interacts with the electrons. We show the relevant operators for both the electron and photon fields and the computation of the corresponding Feynman propagator,S-matrix, and scattering cross section, taking into account the influence of the changeable vacuum field. Correction terms in the computedS-matrix and scattering cross section manifest the influence of the changeable vacuum field. We analyze an example for low-energy scattering of the influence of the changeable vacuum field upon the scattering cross section.     

Four loop anomalous dimensions of gradient operators in {\phi^4} theory

We compute the anomalous dimensions of a set of composite operators which involve derivatives at four loops in in theory as a function of the operator moment . These operators are similar to the twist-2 operators which arise in QCD in the operator product expansion in deep inelastic scattering. By regarding their inverse Mellin transform as being equivalent to the DGLAP splitting functions we explore to what extent taking a restricted set of operator moments can give a good approximation to the exact four loop result. Received: 12 May 1997 / Published online: 20 February 1998     

Spectral Asymptotics for Perturbed Spherical Schrödinger Operators and Applications to Quantum Scattering

We find the high energy asymptotics for the singular Weyl–Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrödinger operators (also known as Bessel operators).We apply this result to establish an improved local Borg–Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.     

Two-cluster scattering ofN charged particles

We define geometrically two-cluster scattering states by their asymptotic space-time behavior. We show that these subspaces coincide with the ranges of the two-cluster wave operators, or modified wave operators if both clusters are charged. In particular this proves asymptotic completeness and absence of a singular continuous spectrum of the Hamiltonian below the lowest three-body threshold.     

Form factors from free fermionic Fock fields,the Federbush model

By representing the field content as well as the particle creation operators in terms of fermionic Fock operators, we compute the corresponding matrix elements of the Federbush model. Only when these matrix elements satisfy the form factor consistency equations involving anyonic factors of local commutativity, the corresponding operators are local. We carry out the ultraviolet limit, analyse the momentum space cluster properties and demonstrate how the Federbush model can be obtained from the SU(3)₃-homogeneous sine-Gordon model. We propose a new class of Lagrangians which constitute a generalization of the Federbush model in a Lie algebraic fashion. We evaluate the associated scattering matrices from first principles, which can alternatively also be obtained in a certain limit of the homogeneous sine-Gordon models.     

The virtual Compton amplitude in the generalized Bjorken region: Twist-2 contributions

A systematic derivation is presented of the twist-2 anomalous dimensions of the general quark and gluon light-ray operators in the generalized Bjorken region in leading order both for unpolarized and polarized scattering. Various representations of the anomalous dimensions are derived in the non-local and local light cone expansion and their properties are discussed in detail. Evolution equations for these operators are derived using different representations. General two- and single-variable evolution equations are presented for the expectation values of these operators for non-forward scattering. The Compton amplitude is calculated in terms of these distribution amplitudes. In the limit of forward scattering a new derivation of the integral relations between the twist-2 contributions to the structure functions is given. Special limiting cases which are derived from the general relations are discussed, as the forward case, near-forward scattering, and vacuum-meson transition. Solutions of the two-variable evolution equations for non-forward scattering are presented.     

Wilsonian RG and redundant operators in nonrelativistic effective field theory

In a Wilsonian renormalization group (RG) analysis, redundant operators, which may be eliminated by using field redefinitions, emerge naturally. It is therefore important to include them. We consider a nonrelativistic effective theory (the so-called “pionless” nuclear effective field theory) as a concrete example and show that the off-shell amplitudes cannot be renormalized if the redundant operators are not included. The relation between the theories with and without such redundant operators is established in the low-energy expansion. We perform a Wilsonian RG analysis for the off-shell scattering amplitude in the theory with the redundant operator.     

On the connection between the inverse transform method and the exact quantum eigenstates

The “inverse scattering transformation”, which has been used to solve certain non-linear field theories classically, is discussed in the context of the quantized version of these theories. In particular we consider the non-linear Schrödinger equation and the massive Thirring model. We find that certain Jost functions of the associated scattering problem lead already, in quantizing the theory, to creation operators for the exact eigenstates of the corresponding Hamiltonians.