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The Lax–Phillips scattering approach and singular perturbation of Schrödinger operator homogeneous with respect to scaling transformations. (English) Zbl 1101.47007

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.

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

Citations:

Zbl 1100.47521
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