A result on the scattering theory for first order systems with long-range perturbations (Q1113018)

The author studies the differential equation \(D_ tu=\Lambda u\), where u is a \(C^ m\)-valued function of \(x\in R^ n\), \(D_ t=(1/i)(\partial /\partial t),\) \(\Lambda =E(x)^{-1/2}\sum^{n}_{j=1}A_ jD_ jE(x)^{-1/2},\) each \(A_ j\) is an m by m constant hermitian matrix, and E(x) is a continuous m by m hermitian matrix valued function with \(0<c_ 1I\leq E(x)\leq c_ 2I\). \(\Lambda\) can be extended to a selfadjoint operator on \({\mathcal H}=L^ 2(R^ n)\). \(\Lambda^ 0=\sum^{n}_{j=1}A_ jD_ j\) can also be extended to a selfadjoint operator on \({\mathcal H}\). In the present paper \(\Lambda\) is regarded as a perturbation of \(\Lambda^ 0\). The main result is the existence of a modified wave operator for \(\Lambda^ 0\) and \(\Lambda\) under the assumption that the perturbation is long-range, i.e., (1) \(E(x)\in C^{\infty}(R^ n)\) and (2) \(| \partial_ x^{\alpha}(E(x)- I)| \leq (1+| x|)^{-\delta -| \alpha |}\) for \(0<\delta \leq 1\) and \(| \alpha | \geq 0\). (In the case of a short- range perturbation, i.e., \(\delta >1\), it is already known that, for a wide class of \(\Lambda^ 0\), the wave operators exist and are complete.) The theory of the paper is applied to the Maxwell equation in crystals.