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\(L^p\)-\(L^{\acute p}\) 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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