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

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.

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

We prove holomorphy of the scattering matrix at fixed energy with respect to c~2 for abstract Dirac operators. Relativistic corrections of order c~2 to the nonrelativistic limit scattering matrix (associated with an abstract Pauli Hamiltonian) are explicitly determined. As applications of our abstract approach we discuss concrete realizations of the Dirac operator in one and three dimensions and explicitly compute relativistic corrections of order c~2 of the reflection and transmission coefficients in one dimension and of the scattering matrix in three dimensions. Moreover, we give a comparison between our approach and the firstorder relativistic corrections according to Foldy-Wouthuysen scattering theory and show complete agreement of the two methods.

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 .

Classical limit for the scattering of Dirac particles in a magnetic field

Journal of Physics A: Mathematical and General, 2003

We present a relativistic quantum calculation at first order in perturbation theory of the differential cross section for a Dirac particle scattered by a solenoidal magnetic field. The resulting cross section is symmetric in the scattering angle as those obtained by Aharonov and Bohm (AB) in the string limit and by Landau and Lifshitz (LL) for the non relativistic case. We show that taking pr 0 |sin (θ/2)| /h ≪ 1 in our expression of the differential cross section it reduces to the one reported by AB, and if additionally we assume θ ≪ 1 our result becomes the one obtained by LL. However, these limits are explicitly singular in h as opposed to our initial result. We analyze the singular behavior inh and show that the perturbative Planck's limit (h → 0) is consistent, contrarily to those of the AB and LL expressions. We also discuss the scattering in a uniform and constant magnetic field, which resembles some features of QCD.

Resonances of the Dirac Hamiltonian in the Non Relativistic Limit

Annales Henri Poincare, 2001

For a Dirac operator in \( {\Bbb R}^3 \) , with an electric potential behaving at infinity like a power of |x|, we prove the existence of resonances and we study, when \( c \rightarrow + \infty \) , the asymptotic expansion of their real part,and an estimation of their imaginary part, generalizing an old result of Titchmarsh.

High-energy and smoothness asymptotic expansion of the scattering amplitude

Journal of Functional Analysis, 2003

We find an explicit expression for the kernel of the scattering matrix for the Schrödinger operator containing at high energies all terms of power order. It turns out that the same expression gives a complete description of the diagonal singularities of the kernel in the angular variables. The formula obtained is in some sense universal since it applies both to short-and long-range electric as well as magnetic potentials.

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.

General boundary conditions for a Dirac particle in a box and their non-relativistic limits

The most general relativistic boundary conditions (BCs) for a 'free' Dirac particle in a one-dimensional box are discussed. It is verified that in the Weyl representation there is only one family of BCs, labelled with four parameters. This family splits into three sub-families in the Dirac representation. The energy eigenvalues as well as the corresponding non-relativistic limits of all these results are obtained. The BCs which are symmetric under space inversion P and those which are CP T invariant for a particle confined in a box, are singled out. † for example, by cancelling only parts of the electric and magnetic fields at the boundary, it is found that a non-trivial solution exists (stationary waves).