Holomorphy of the scattering matrix with respect toc− 2 for Dirac operators and an explicit treatment of relativistic corrections (original) (raw)

Relativistic corrections for the scattering matrix for spherically symmetric potentials

Reports on Mathematical Physics, 2001

We use the framework for the nonrelativistic limit of scattering theory for abstract Dirac operators developed in [5] to prove holomorphy of the scattering matrix at fixed energy with respect to c −2 for Dirac operators with spherically symmetric potentials. Relativistic corrections of order c −2 to the nonrelativistic limit partial wave scattering matrix are explicitly determined.

Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits

Arkiv för Matematik, 1999

In this paper we prove several results for the scattering phase (spectral shift function) related with perturbations of the electromagnetic field for the Dirac operator in the Euclidean space. Many accurate results are now available for perturbations of the Schr5dinger operator, in the high energy regime or in the semi-classical regime. Here we extend these results to the Dirac operator. There are several technical problems to overcome because the Dirac operator is a system, its symbol is a 4 x 4 matrix, and its continuous spectrum has positive and negative values. We show that we can separate positive and negative energies to prove high energy asymptotic expansion and we construct a semi-classical Foldy Wouthuysen transformation in the semi-classical case. We also prove an asymptotic expansion for the scattering phase when the speed of light tends to infinity (non-relativistic limit).

Relativistic scattering with a spatially dependent effective mass in the Dirac equation

2007

We formulate an algebraic relativistic method of scattering for systems with spatially dependent mass based on the J-matrix method. The reference Hamiltonian is the threedimensional Dirac Hamiltonian but with a mass that is position-dependent and having a constant asymptotic limit. Additionally, this effective mass distribution is locally represented in a finite dimensional function subspace. The spinor couples to spherically symmetric vector and pseudo scalar potentials that are short-range such that they are accurately represented by their matrix elements in the same finite dimensional subspace. We calculate the relativistic phase shift as a function of energy for a given configuration and study the effect of spatial variation of the mass on the energy resonance structure.

Integral identities and bounds for scattering calculations and the Dirac formalism

Physical Review A, 1999

Integral identities that hold between "desired" and "comparison" solutions of the radial Dirac equations for scattering precesses are considered. Applications of these identities are discussed, particularly the determination of bounds to variational calculations of K-matrix elements. Relativistic effects in atomic scattering, and indeed in many quantum scattering processes, can usually be treated as small corrections, and handled by perturbation theory with respect to the non-relativistic (Schrödinger) results. However, for some processes, such as the scattering of electrons by atoms and molecules, and for many nuclear scattering processes, it is often convenient or even necessary to use the Dirac equation directly. This is because the relativistic kinematics and spin effects are then automatically "built in" (see, for example, ref. [1-3] and citations therein). In addition to this, relativistic corrections to the dynamics (i.e. corrections to the static potential) may need to be taken into account.

High-energy behavior of the scattering amplitude for a Dirac operator

Publications of the Research Institute for Mathematical Sciences, 1995

We study the high-energy behavior of the scattering amplitude and the total scattering cross section for a Dirac operator with a 4 X 4 matrix-valued potential. Moreover, in the electromagnetic case, it is shown that the electric potential and the magnetic field can be reconstructed from the high-energy behavior of the scattering amplitude. The study of the high-energy behavior of the resolvent estimates is crucial for our proof.

On algebraic models of relativistic scattering

Journal of Physics A, 2008

In this paper we develop an algebraic technique for building relativistic models in the framework of the direct-interaction theories. The interacting mass operator M in the Bakamjian-Thomas construction is related to a quadratic Casimir operator C of non-compact group G. As a consequence the S matrix can be gained from an intertwining relation between Weyl-equivalent representation of G. The method is illustrated by explicit application to a model with SO(3, 1) dynamical symmetry.

Scattering amplitude for dirac operators

Communications in Partial Differential Equations, 1999

hl.Taylor has constructed an approximate diagonalization of a system of pseudodifferential operators under the assumption that the matrix of principal symbols is diagonalizable in a C" way. Helffer and Sjostrand [9] have given the analogue of Taylor's construction in the semiclassical case. For some systems, like t h system of h9axwell equations ([13]), the hypothesis of global Cw diagonalizability cannot be met, for example for topological reasons. and it seems useful to have a variant of the above result which does not need this hypothesis, and gives approximate projectors rather than an approximate diagonalization. We present such a result in section 2 below, with a proof which differs from those in [17] and .

On the Dirac Scattering Problem

We consider a method of solving the Dirac scattering problem based on an approach previously used by the authors to solve the Schrödinger scattering problem to develop a conditional exact scattering solution and an unconditional series solution. We transform the Dirac scattering problem into a form that facilitates a solution based on the relativistic Lippmann-Schwinger equation using the relativistic Green's function that is transcendental in terms of the scattered field. Using the Dirac operator, this solution is transformed further to yield a modified relativistic Lippmann-Schwinger equation that is also transcendental in terms of the scattered field. This modified solution facilitates a condition under which the solution for the scattered field is exact. Further, by exploiting the simultaneity of the two solutions available, we show that is possible to define an exact (non-conditional) series solution to the problem.

On the spectral theory of Dirac type hamiltonians

JOURNAL OF OPERATOR THEORY

We introduce a class of matrix valued pseudo-dierentia l opera- tors that admit scalar locally conjugate operators (in the sense of E. Mourre) and we give a general method of study of singular perturbations of such oper- ators. In particular, we develop the spectral and scattering theory for a class of hamiltonians which contains the Dirac operators with arbitrary Coulomb singularities.