The continuity in \(L^{p}\) spaces and Sobolev spaces of the wave operators associated to Schrödinger Hamiltonians acting on spaces of even dimension is proved. Previously, similar properties for operators acting on spaces of odd dimension were proved in K. Yajima’s paper [J. Math. Sci., Tokyo 13, No. 1, 43–93 (2006; Zbl 1115.35094)].
We give below the more detailed description of the results which are proved. Always in what follows \(m\geq 6\) is an even number. Let \(H_{0}\) be the free Schrödinger operator, \(H_{0} = -\Delta\) and \(H=H_{0}+V\), where \(| V(x) | \leq C(1+ | x | ^{2})^{-\delta}\) for some \(\delta > 1\) and some \(C>0\), and the Fourier transform of \((1+ | x | ^{2})^{\sigma} V\) is in \(L^{m_{\star}}\) for \(\sigma > 1 / m_{\star} = (m-2)/ m-1\). The operator \(H\) is said to be of exceptional type if the equation \(-\Delta \phi + V\phi = 0\) has at least one non-trivial solution which satisfies \( | \phi(x) |\leq C(1+ | x | ^{2})^{(2-m)/2}\), and is said to be of generic type otherwise.
One proves that if \(H\) is of generic type and if, in addition, \(| V(x) | \leq C(1+ | x | ^{2})^{-(m+2+\varepsilon)/2}\) for some \(\varepsilon > 0\) and some \(C>0\), then the wave operators associated to \(H\) and \(H_{0}\) can be extended to bounded operators in the Sobolev spaces of order \(k\), \(W^{k,p}(\mathbb{R}^{m})\) for \(0\leq k\leq 2\) and \(1 < p < \infty\) and to continuous operators in \(L^{1}(\mathbb{R}^{m})\) and in \(L^{\infty}(\mathbb{R}^{m)}\). If \(H\) is of exceptional type and if \(m=6\) and \(| V(x) | \leq C(1+ | x | ^{2})^{-(m+4+\varepsilon)/2}\) or if \(m\geq 8\) and \(| V(x) | \leq C(1+ | x | ^{2})^{-(m+3+\varepsilon)/2}\) for some \(\varepsilon > 0\) and some \(C>0\), then the wave operators can be extended to bounded operators in \(W^{k,p}(\mathbb{R}^{m})\) for \(m /( m-2) < p < m / 2\) and \(0 \leq k\leq 2\). Continuity properties in Sobolev spaces of higher order are also proved for potentials \(V\) with bounded derivatives.
The proofs are based, as in the odd case, on the stationary representation of the wave operators. The high energy and the low energy parts of the wave operators are studied separately.