Kuzhel, Sergei; Moskalyova, Uliya The Lax–Phillips scattering approach and singular perturbation of Schrödinger operator homogeneous with respect to scaling transformations. (English) Zbl 1101.47007 J. Math. Kyoto Univ. 45, No. 2, 265-286 (2005). In [S. Kuzhel, Ukr.Mat.Zh.55, No. 5, 621–630 (2003); translation in Ukr.Math.J.55, No. 5, 749–760 (2003; Zbl 1100.47521)], the first author gave a condition on the positive selfadjoint operator \(L\) equivalent to the existence of outgoing and incoming spaces, in the sense of Lax and Phillips, for the general wave equation \(u_{tt}=-Lu\). In the paper under review, this result is applied to the case of singular perturbations of the Laplacian. Namely, the authors prove that the aforementioned result is applicable to a class of operators of the form \(-\Delta +V\), \(V:W^{2}_{2}(\mathbb{R}^{3})\rightarrow W^{-2}_{2}(\mathbb{R}^{3})\) (here, \(W^{2}_{2}\) is the Sobolev space of order 2) homogeneous with respect to the scaling transformations \((G(t)f)(x)=t^{3/2}f(tx)\). Consequently, the existence and unitarity of the wave operators associated to such perturbations and a description of the scattering matrix are obtained. Reviewer: Mihai Pascu (Bucureşti) Cited in 2 Documents MSC: 47A40 Scattering theory of linear operators 47B25 Linear symmetric and selfadjoint operators (unbounded) 35P25 Scattering theory for PDEs 81U05 \(2\)-body potential quantum scattering theory Keywords:Lax-Phillips scattering theory; Schrödinger operator; singular perturbations; self-adjoint extensions; wave operators Citations:Zbl 1100.47521 PDFBibTeX XMLCite \textit{S. Kuzhel} and \textit{U. Moskalyova}, J. Math. Kyoto Univ. 45, No. 2, 265--286 (2005; Zbl 1101.47007) Full Text: DOI