Scattering theory by Enss' method for operator valued matrices: Dirac operator in an electric field
Existence of the modified wave operators and asymptotic completeness using Enss' method are proved for the operator \(H=H_ 0(P)+W_ S(Q,P)+W_ L(Q,P)I_ m\) on \([L^ 2(R^ n)]^ m\) where (i) \(H_ 0\) is a smooth map on \(R^ n\) taking values in the space of \(m\times m\) Hermitian matrices such that \(\lim_{| \zeta | \to \infty}| \lambda (\zeta)| =\infty\) for each eigenvalue \(\lambda\) (\(\zeta)\) of \(H_ 0(\zeta),\) (ii) \(W_ S(Q,P)\) is a short range pseudo differential operator on \([L^ 2(R^ n)]^ m,\) (iii) \(I_ m\) is the identity \(m\times m\) matrix and (iv) \(W_ L(Q,P)\) is a smooth long range pseudo differential operator on \(L^ 2(R^ n).\) The model includes the Dirac operator on \([L^ 2(R^ 3)]^ 4\) with an electric field.
short range pseudo differential operator
scientific article; zbMATH DE number 3929748