Time dependent representations of the stationary wave operators for ''oscillating'' long-range potentials
Assume that \(V:{\mathbb{R}}^ n\to {\mathbb{R}}\) is a potential that can be split into a short-range part and a long-range term \(V_ L\) which may oscillate in a certain way. Let c be a real number which is specified below. The unique self-adjoint extensions of \(-\Delta +c\) and \(-\Delta +V\) in \(L^ 2({\mathbb{R}}^ n)\) are denoted by \(H_ 0\) and H, respectively, and \(E_ 0(I)\) is the spectral projection of \(H_ 0\) for the interval \(I\subset\subset {\mathbb{R}}\). With the help of functions \(\delta_+\) and \(\delta_-\) which satisfy a certain eikonal inequality, and using the spectral representation of \(H_ o\), the authors introduce identification operators \(J_{\pm}(I)\). They prove that \[ W_{\pm}(t):=e^{itH}J_{\pm}(I)e^{-itH_ 0}E_ 0(I)\quad (t\in {\mathbb{R}}) \] are bounded operators by following a method of \textit{A. P. Calderón} and \textit{R.Vaillancourt} [Proc. Nat. Acad. Sci. USA 69, 1185- 1187 (1972; Zbl 0244.35074)] to show the \(L^ 2\)-boundedness of pseudo- differential operators. Next, using the stationary phase method, they prove that \(W_{\pm}(t)\) converge strongly as \(t\to\pm \infty\) to wave opertors \(U_{\pm}(I)^{\pm}\) the existence and completeness of which the authors had established by means of a time-independent method in an earlier paper [J. Math. Kyoto Univ. 19, 47-70 (1979; Zbl 0414.47013)]. The position of the interval I is determined by the number c which is explicitly given, essentially in terms of the asymptotic behaviour of \[ r \partial V_ L/\partial r+const.\quad V_ L. \] The authors mention that earlier modified wave operators coincide modulo simple unitary operators with the usual Møller wave operators when V is entirely short-range, but that it is an open problem whether this continues to hold for long-range potentials for which the latter exist, for example \(V=V_ L=a \sin br/r.\)
time dependent representations of the stationary wave operators
\(L^ 2\)-boundedness of pseudo-differential operators
scientific article; zbMATH DE number 3876005
scientific article