×

The \(W^{k,p}\)-continuity of wave operators for Schrödinger operators. III: Even dimensional cases \(m \geq 4\). (English) Zbl 0841.47009

Summary: Let \(H= -\Delta+ V(x)\) be the Schrödinger operator on \(\mathbb{R}^m\), \(m\geq 3\). We show that the wave operators \(W_\pm= \lim_{t\to \pm\infty} e^{itH}\) \(e^{- itH_0}\), \(H_0= -\Delta\), are bounded in Sobolev spaces \(W^{k, p}(\mathbb{R}^m)\), \(1\leq p\leq \infty\), \(k= 0, 1,\dots, \ell\), if \(V\) satisfies \(|D^\alpha V(y)|_{L^{p_0}(|x- y|\leq 1)}\leq C(1+ |x|)^{- \delta}\) for \(\delta> (3m/2)+ 1\), \(p_0> m/ 2\) and \(|\alpha|\leq \ell+ \ell_0\), where \(\ell_0= 0\) if \(m= 3\) and \(\ell_0= [(m- 1)/2 ]\) if \(m\geq 4\), \([\sigma]\) is the integral part of \(\sigma\). This result generalizes the author’s previous result which appears in J. Math. Soc. Jap. 47, No. 3, 551-581 (1995), where the theorem is proved for the odd dimensional cases \(m\geq 3\) and several applications such as \(L^p\)-decay of solutions of the Cauchy problems for time-dependent Schrödinger equations and wave equations with potentials, and the \(L^p\)-boundedness of Fourier multiplier in generalized eigenfunction expansions are given.
[For part II see Lect. Notes Pure Appl. Math. 161, 287-300 (1994; Zbl 0820.35114)].

MSC:

47A40 Scattering theory of linear operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35P25 Scattering theory for PDEs
81Uxx Quantum scattering theory
35Q40 PDEs in connection with quantum mechanics

Citations:

Zbl 0820.35114
PDF BibTeX XML Cite