Asymptotic dynamics and spectral analysis for the Schrödinger operator with weakly accelerating potential (Q1263739)

The author proves unitary equivalence of the infinitesimal translation operator and the self-adjoint Schrödinger operator \(H=-\partial^ 2_ x+v(x)\) in \(L_ 2({\mathbb{R}}_+)\) with a weakly accelerating potential v(x) that for sufficiently large x satisfies such conditions \[ -v_- x^{2\alpha}\leq v(x)\leq -v_+x^{2\alpha},\quad 0<\alpha <1,\quad v_+>0, \] \[ | v'(x)| \leq v_ 1x^{(5\alpha_ 1-1)},\quad | v''(x)| \leq v_ 2x^{-1+3\alpha_ 1};\quad \alpha_ 1<\alpha. \] A function \(f(t)=\exp (-iHt)f(0)\) is proved to be asymptotically a wave package that moves uniformly to infinity with a steady form for \(\alpha >1/3\) and a changing one for \(\alpha\leq 1/3\).