Asymptotic completeness in singular potential scattering derived by geometric methods

A general two-space criterion for the asymptotic completeness is derived. It is applied to the potential scattering with infinitely high potentials over unbounded regions.     

Asymptotic completeness for a quantum particle in a Markovian short range potential

Absence of bound states and asymptotic completeness are proven for a quantum particle in a time dependent random (Markovian) short range potential. Systems with confining potentials are also considered and unboundedness of the energy in time is shown.     

Asymptotic completeness for quantum-mechanical potential scattering: II. Singular and long-range potentials

The geometrical proof for the asymptotic completeness of the wave operators and the absence of a singular continuous spectrum of the Hamiltonian is extended to cover singular potentials where the Hamiltonian is defined as a form sum, and long-range interactions like Coulomb forces.     

Asymptotic completeness in nonrelativistic quantum field theory

We describe some simple quantum field-theoretical models in which asymptotic completeness can be shown.     

Asymptotic completeness for a quark model of meson scattering

We prove asymptotic completeness for the scattering of two non-relativistic spinless mesons, when each meson is composed of a quark and antiquark bound together by a confining potential, so that free quarks may not result from the collision.     

Asymptotic completeness of scattering in the nonlinear Lamb system for nonzero mass

We establish the asymptotic completeness in the nonlinear Lamb system with nonzero mass for hyperbolic stationary states. For the proof, we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point, using the Banach space Inverse Function Theorem and a priori estimates. We give a counterexample showing that the hyperbolicity condition is essential.     

Asymptotic completeness forN-body short-range quantum systems: A new proof

We give an alternative geometrical proof of asymptotic completeness for an arbitrary number of quantum particles interacting through shortrange pair potentials. It relies on an estimate showing that the intercluster motion concentrates asymptotically on classical trajectories.Dedicated to Res Jost and Arthur Wightman     

Eigenchannel method in quantum potential scattering

The eigenchannel method, generalizing the familiar phaseshift method, is formulated for scattering from a Hermitian short range potential. Scattering eigenchannels are defined as eigenstates of some generalized (weighted) operator spectral problem. Eigenvalues of that problem define eigenphaseshifts, the former being the negative of cotangents of the latter. Eigenchannel representations of generalized scattering states, transition operators, and Green operators are constructed. A variational approach to the method is also presented. The general theory is illustrated by applications to scattering of Schrödinger and Dirac particles.     

Asymptotic completeness forN-body Stark Hamiltonians

We prove asymptotic completeness for short- and long-rangeN-body Stark Hamiltonians with local singularities of at most Coulomb type. Our results include the usual models for atoms and molecules.Research partially supported by NSF grant DMS 9307147.     

An example in potential scattering illustrating the breakdown of asymptotic completeness

An example is given of a local spherically symmetric short range potential, such that the wave operators for scattering of a single particle by the potential are not complete. States exist which are asymptotically free att=?∞, but having a non zero probability of absorption into the origin att=+∞.     

On the break-down of completeness of wave operators in potential scattering

We study the Schrödinger operator with a potential that vanishes at infinity but the rate of falloff of the potential depends on the direction. It turns out that for such potentials scattering theory becomes in general multichannel.     

Asymptotic localization and separation and the scattering of quantum states

The concepts of asymptotic localization and separation of quantum states introduced in a previous paper are generalized to the n-dimensional case. An application of these ideas to a simple scattering situation leads to an intuitively pleasing picture of scattering.     

Asymptotic completeness for multi-particle schroedinger Hamiltonians with weak potentials

We show that the non-relativistic quantum mechanicaln-body HamiltoniansT(k)=T+kV andT, the free particle Hamiltonian, are unitarily equivalent in the center of mass system, i.e.,T(k)=W _± (k)TW _± (k) ^–1 fork sufficiently small and real. , a sum ofn(n–1)/2 real pair potentials,V _i, depending on the relative coordinatex _i R ³ of the pairi, whereV _i is required to behave like |x_i|^{– 2 –} as |x _i | and like |x_i|^{– 2 +} as |x _i |0.T(k) is the self-adjoint operator associated with the form sumT+kV. There are no smoothness requirements imposed on theV _i . Furthermore are the wave operators of time dependent scattering theory and are unitary. This result gives a quantitative form of the intuitive argument based on the Heisenberg uncertainty principle that a certain minimum potential well depth and range is needed before a bound state can be formed. This is the best possible long range behavior in the sense that ifkV _i C _i |x _i |^–b , 0<b2 for |x _i |>R _i (0<R _i <) and allC _i are negative thenT(k) has discrete eigenvalues andW _±(k) are not unitary.     

Classical and quantum scattering by a Coulomb potential

For relativistic energies the small-angle classical cross section for scattering on a Coulomb potential agrees with the first Born approximation for quantum cross section for scalar particle only in the leading term. The disagreement in other terms can be avoided if the sum of all corrections to the first Born approximation for large enough Coulomb charge contains the classical terms which are independent of that charge. The difference in classical and quantum cross sections may be partly attributed to the fact that the relativistic quantum particle can rush through the field without interaction. We expect that smaller impact parameters and spin facilitate this effect. The text was submitted by the authors in English.     

Stochastic interpretation of potential scattering in quantum mechanics

We generalize Shucker's result on the asymptotically-classical behaviour of the quantum system with zero potential in Nelson's stochastic interpretation of quantum mechanics to the case of the one-dimensional potential scattering.     

Asymptotic completeness for a new class of Stark effect Hamiltonians

Existence and completeness of the wave operators is shown for the Stark effect Hamiltonian in one dimension with a potentialV =W, whereW is a bounded function with four bounded derivatives. This class of potentials include some almost periodic functions and periodic functions with average zero over a period (Stark-Wannier Hamiltonians). In the last section we discuss classical particle scattering for the same class of potentials.Partially supported by NSF-grant DMS-8401748     

Mixed quantum mechanical periodic and potential wall problem

We consider the Schrödinger equation with an even-square integrable potential of period one on the negative real axis and a wall potential of heighta > 0 on the positive real axis. The spectrum of this Schrödinger equation is determined and it is proved that bounded solutions never exist if the energyE < a is lying in a gap of the periodic spectrum.     

Effective potential approach to the quantum scattering of solitons

Systems of solitons are approximately described in terms of a finite number of “effective degrees of freedom” interacting via “effective potentials”. These are reconstructed, principally from knowledge of solutions to the classical field equations, by a procedure involving the sometric mapping of a sector of the field theoretical Hilbert space onto the Hilbert space of non-relativistic point particles. The quantum dynamics of solitons is then approximately formulated and solved in terms of ordinary Schrödinger-type equations. As an interesting exercise, our method is applied to an analysis of soliton-antisolotion binding and scattering in the sine-Gordon model. With the exclusion of exceptional values of the coupling constant, corresponding to solutions of an eigenvalue equation, backward scattering is found to occur near threshold and to decay exponentially with the centre-of-mass energy. One of the exceptional, reflectionless sine-Gordon model is, not surprisingly, found to correspond to vanishing coupling in the massive Thirring model.