Structure of the scattering operator for time-periodic Schrödinger equations
Let \[ i(\partial /\partial t)\phi (t,x)=(-\Delta +V(t,x))\Phi (t,x),\quad \Phi (t,\cdot)\in L^ 2({\mathbb{R}}^ n), \] \[ V(t+\omega,x)=V(t,x)\in {\mathbb{R}}\quad (t\in {\mathbb{R}},\quad x\in {\mathbb{R}}^ n) \] be a time-dependent Schrödinger equation with period \(\omega\). The structure of the scattering operator \(S(s)=W_+(s)*W_-(s)\) is analyzed, where \[ W_{\pm}(s)=s-\lim_{t\to \pm \infty} U(s,t)e^{- i(t-s)H_ 0},H_ 0=-\Delta, \] U being the corresponding unitary evolution operator. Assume for the potential the following: For some \(p>n\), \(\alpha >1/2\), \(\beta >0\), \(t\to (1+| x|^ 2)^{\alpha}V(t,x)\) is an \((L^ p({\mathbb{R}}^ n)+L^{\infty}({\mathbb{R}}^ n))\)-valued \(C^{1+\beta}\)-function. Let \(\{e^{-i\omega \lambda_ j}\}_{j\in {\mathbb{N}}}\) denote the eigenvalues of \(U(s+\omega,s)\), \(E:=\{(2\pi /\omega)\mu +\lambda_ j:\) \(\mu\in {\mathbb{Z}},j\in {\mathbb{N}}\}\cup (2\pi /\omega)Z\). Then the main result is: If J is a compact subset of \({\mathbb{R}}\) such that \(J\cap E=\emptyset\) and \(\epsilon >\beta\) is a positive constant, then \[ \| P_{(\lambda:\lambda >E)}(H_ 0)S(s)P_ j(H_ 0)\| <CE_ 1^{-(14+\epsilon)}\quad (E_ 1>0), \] where \(\{P_{\Omega}(H_ 0\}\) is the spectral measure of \(H_ 0\).
scientific article; zbMATH DE number 3964535