The \(W^{k,p}\)-continuity of wave operators for Schrödinger operators. (English) Zbl 0837.35039

Yajima, Kenji

J. Math. Soc. Japan 47, No. 3, 551-581 (1995).

The author investigates wave operators \(W_\pm(H, H_0)\), where \(H_0\) is Laplace operator in \(L^2(\mathbb{R}^m)\), \(H= H_0+ V\), and \(V\) is the multiplication operator. It is shown that under suitable conditions on \(V\), that \(W_\pm\) are bounded operators in the Sobolev spaces \(W^{k, p}(\mathbb{R}^m)\) for any \(1\leq p\leq \infty\), \(k= 0,\dots,\ell\). In particular, this result implies that the functions \(f(H_0)\) and \(f(H) P_c(H)\), \(P_c\) being the orthogonal projection onto the continuous spectral subspace of \(H\), have equivalent operator norms from \(W^{k,p}(\mathbb{R}^m)\) to \(W^{n, q}(\mathbb{R}^m)\) for any \(1\leq p\), \(q\leq \infty\), \(k, n= 0,1,\dots,\ell\).

Reviewer: M.Perelmuter (Kiev)

MSC:

35J10	Schrödinger operator, Schrödinger equation
47F05	General theory of partial differential operators

Keywords:

spectral subspace; equivalent operator norms