## $$L^p$$-$$L^{\acute p}$$ estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential.(English)Zbl 0943.34070

It is proved that the absolutely continuous part of the $$1$$-D Schrödinger operator on the line generates a unitary group satisfying the $$L^p$$-to-$$L^{p/(p-1)}$$ estimate $$\|e^{-iHt}P_c\|\leq C t^{1/p-1/2}$$ for $$t>0$$ whenever the potential $$V$$ satisfies $$(1+|\cdot|)^\gamma V\in L^1({\mathbb{R}})$$ for $$\gamma>3/2$$ (no half-bound state at zero energy) or for $$\gamma>5/2$$ (half-bound state at zero energy). Under these assumptions and for $$H$$ having no negative eigenvalues, the nonlinear Schrödinger equation $i{{\partial}\over{\partial t}}u(t,x)=Hu+f(|u|){{u}\over{|u|}},$ where $$f$$ is real-valued and $$C^1$$, $$f(0)=0$$ and $$|f^\prime(\mu)|\leq C|\mu|^{p-1}$$ for some $$p\geq 5$$, is shown to be uniquely solvable for small remote past data and to admit a scattering operator $$S$$ that uniquely determines the potential $$V$$.

### MSC:

 34L25 Scattering theory, inverse scattering involving ordinary differential operators 35Q55 NLS equations (nonlinear Schrödinger equations) 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 81U40 Inverse scattering problems in quantum theory
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