##
**The \(L^p\)-continuity of wave operators for one dimensional Schrödinger operators.**
*(English)*
Zbl 0976.34071

The authors consider the one-dimensional wave operator \(W_\pm=s -\lim_{t\to\pm\infty} e^{itH}e^{-itHo}u\) for the Schrödinger operators \(H_0=-d^2/dx^2\) and \(H=H_0+ V(x)\), with \(x\in\mathbb{R}\). (At this point it would be nice if the authors could mention to the uninitiated reader what kind of symbol \(s\) is.)

\(V\) is as usual the potential, which is called generic if the Wronskian determinant \[ [f_+(x,0), f_-(x,0)]\neq 0, \] where \(f_\pm(x,k)\) are solutions to the equation: \(- f^n+Vf =k^2f\). The authors introduce the weighted \(L^1_\gamma\) Banach space whose elements are the functions \(\{u: \int_{\mathbb{R}}(1+|x|)^\gamma|u(x)|dx <\infty\}\). They prove the following theorem: If \(V\in L^1_3(\mathbb{R})\) and it is of the generic type, or else if \(V\in L^1_4(\mathbb{R})\) and is of exceptional type (i.e. not generic) and \(V'\in L^1_2(\mathbb{R})\) then \(W_\pm\) is a bounded operator in \(L^p(\mathbb{R})\) for any \(p\) satisfying \(1 <p< \infty\), and moreover \(W_\pm u\) defined on \(L^p(\mathbb{R})\cap L^2(\mathbb{R})\) has a unique extension bounded in \(L^p(\mathbb{R})\). Some technical theorems and lemmas follow, but the main purpose of the paper is the proof of this important theorem.

A similar result, but with a different technique of proof and with somewhat weaker assumptions, obtained by R. Weder, is mentioned by the authors, and is listed in their references. Their work and Widen’s work were completed at almost the same time in 1999. For mathematicians working in this area a comparison of the techniques should be of interest.

\(V\) is as usual the potential, which is called generic if the Wronskian determinant \[ [f_+(x,0), f_-(x,0)]\neq 0, \] where \(f_\pm(x,k)\) are solutions to the equation: \(- f^n+Vf =k^2f\). The authors introduce the weighted \(L^1_\gamma\) Banach space whose elements are the functions \(\{u: \int_{\mathbb{R}}(1+|x|)^\gamma|u(x)|dx <\infty\}\). They prove the following theorem: If \(V\in L^1_3(\mathbb{R})\) and it is of the generic type, or else if \(V\in L^1_4(\mathbb{R})\) and is of exceptional type (i.e. not generic) and \(V'\in L^1_2(\mathbb{R})\) then \(W_\pm\) is a bounded operator in \(L^p(\mathbb{R})\) for any \(p\) satisfying \(1 <p< \infty\), and moreover \(W_\pm u\) defined on \(L^p(\mathbb{R})\cap L^2(\mathbb{R})\) has a unique extension bounded in \(L^p(\mathbb{R})\). Some technical theorems and lemmas follow, but the main purpose of the paper is the proof of this important theorem.

A similar result, but with a different technique of proof and with somewhat weaker assumptions, obtained by R. Weder, is mentioned by the authors, and is listed in their references. Their work and Widen’s work were completed at almost the same time in 1999. For mathematicians working in this area a comparison of the techniques should be of interest.

Reviewer: Vadim Komkov (Florida)

### MSC:

34L25 | Scattering theory, inverse scattering involving ordinary differential operators |

81U05 | \(2\)-body potential quantum scattering theory |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |