Moller Wave Operators Research Articles

Trace class properties of the non homogeneous linear Vlasov–Poisson equation in dimension 1+1

We consider the abstract scattering structure of the non homogeneous linearized Vlasov–Poisson equations from the viewpoint of trace class properties which are emblematic of the abstract scattering theory [13, 14, 15, 19]. In dimension 1+1, we derive an original reformulation which is trace class. It yields the existence of the Moller wave operators. The non homogeneous background electric field is periodic with 4 + \varepsilon bounded derivatives.

Generalizations of Menchov–Rademacher Theorem and Existence of Wave Operators in Schrödinger Evolution

AbstractWe obtain generalizations of the classical Menchov–Rademacher theorem to the case of continuous orthogonal systems. These results are applied to show the existence of Moller wave operators in Schrödinger evolution.

DISCRETE QUANTUM SCATTERING THEORY

We formulate quantum scattering theory in terms of a discrete L2-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: S- and T-matrices, resolvent operators, and Moller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the T-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators.

Scattering and Wave Operators for One-Dimensional Schr�dinger Operators with Slowly Decaying Nonsmooth Potentials

We prove existence of modified wave operators for one-dimensional Schrodinger equations with potential in \(L^p (\mathbb{R}),p < 2.\) If in addition the potential is conditionally integrable, then the usual Moller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential.

On the Point-Particle (Newtonian) Limit¶of the Non-Linear Hartree Equation

We consider the nonlinear Hartree equation describing the dynamics of weakly interacting non-relativistic Bosons. We show that a nonlinear Moller wave operator describing the scattering of a soliton and a wave can be defined. We also consider the dynamics of a soliton in a slowly varying background potential W(ɛx). We prove that the soliton decomposes into a soliton plus a scattering wave (radiation) up to times of order ɛ−1. To leading order, the center of the soliton follows the trajectory of a classical particle in the potential W(ɛx).

Wave functions for continuum states of charged fragments.

Briggs's representation [Phys. Rev. A 41, 539 (1990)] of the Moller wave operator for multiparticle wave functions is applied to charged fragments using a limiting procedure to correctly account for the slow decrease of Coulomb interactions with distance. Approximate wave functions used to model ([ital e],2[ital e]) angular correlation measurments are obtained. Computed and measured angular correlations are compared to clarify the region of applicability of two approximations.

Computation of a long-time evolution in a Schrödinger system.

We compare different techniques for the computation of a long-time evolution and the S matrix in a Schr\"odinger system. As an application we consider a two-nucleon system interacting via the Yamaguchi potential. We suggest computation of the time evolution for a very short time using Pad\'e approximants, the long-time evolution being obtained by iterative squaring. Within the technique of strong approximation of Moller wave operators (SAM) we compare our calculation with computation of the time evolution in the eigenrepresentation of the Hamiltonian and with the standard Lippmann-Schwinger solution for the S matrix. We find numerical agreement between these alternative methods for time-evolution computation up to half the number of digits of internal machine precision, and fairly rapid convergence of both techniques towards the Lippmann-Schwinger solution.

Algebraic reduction and approximate intertwining relation in the strong approximation of Moller wave operators approach.

Nonrelativistic quantum scattering processes can be calculated from strong approximations of Moller wave operators, based on finite rank approximations of the full and asymptotic Hamiltonians. Firstly, similarly to Faddeev-type integral equations, we show that this approach also reduces a three-body system with separable two-body interactions to an effective two-body system. In the simplest case of an s-wave interaction without tensor forces, it reduces to a one-dimensional equation after partial wave decomposition. Secondly, we show that the intertwining relation is approximately valid in this approach and holds in the limit where the approximation parameters tend to infinity. We give a proof for the two-body system with short range interactions and for the long range Coulomb interaction. Implications for practical calculations are discussed.

Coulomb scattering of wave packets

Advantages are pointed out in studying S-matrix elements between wave packets when long range Coulomb forces are involved. A straightforward method is presented to calculate such matrix elements in few-body systems. It is based on a strong approximation of the Moller wave operators. For the two-body system the method is tested for (a) short range, (b) Coulomb, and (c) short range plus Coulomb potentials, and is found in all cases to converge to the reference solution. As an application, d+p breakup cross sections are calculated close to the threshold, including Coulomb forces properly.

Wave operators for atomic photo-ionisation

The time development of a system consisting of an atom in an external electromagnetic field with fixed propagation direction is studied (vector potential A(x,t)=A(x3-ct)). It is shown that the Moller wave-operators, relevant for photo-ionisation, exist, provided the components Asigma (u) of A decay as mod u mod -1- epsilon , epsilon >0, for large argument u. Since A(x,t) is not necessarily spatially homogeneous, the above results can serve as a starting point for an investigation of field-gradient effects on photo-ionisation processes.

Phase shifts and the modified and møller wave operators

The Schrodinger operator H=−Δ+V is considered when the potential V is central, oscillating and possibly unbounded. An eigenfunction expansion for H is derived. Also, the phase shift of the generalized eigenfunctions of H is computed. The expansion and shift are, together, applied to the study of the wave operators of scattering theory. The modified wave operators, for H and H0=−Δ are shown to exist and be complete and a condition on V is derived which is necessary and sufficient for the Moller wave operators to exist and be complete. Spectral properties of H are also discussed.

A new proof of the invariance principle in scattering theory

For free and interacting Hamiltonians, H 0 and H = H 0 + V( r) acting in L 2(R 3, dx) with V( r) a radial potential satisfying certain technical conditions, and for ϕ a real function on R with ϕ′ > 0 except on a discrete set, we prove that the Moller wave operators Ω± = strong limit e itϕ(H) e −itϕ(H 0) exist and are independent of ϕ. The scattering operator S = (Ω +)∗Ω − is shown to be unitary. Our proof utilizes time independent methods (eigenfunction expansions) and is effective in cases not previously analyzed, e.g. V(r) = sin r r and many others.

The relationship between formal scattering theory and the density matrix

A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Moller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.