Absolute continuity of the Floquet spectrum for a nonlinearly forced harmonic oscillator (Q5940553)

The authors consider the time-dependent Schrödinger equation of the form \[ i{\partial u\over\partial t}= -\textstyle{{1\over 2}}\Delta u+ \textstyle{{1\over 2}} x^2 u+ 2\varepsilon(\sin t)x_1 u+\mu V(t,x) u,\tag{1} \] where \(V(t,x)\) is real-valued smooth function of \((t,x)\), \(2\pi\)-periodic with respect to \(t\), and \[ |\partial^\alpha_x V(t,x)|\leq C_\alpha,\quad|\alpha|\geq 1. \] Assuming additionally \(|\mu|\sup_{t,x}|\partial x_1 V(t,x)|\leq \varepsilon\), the authors show that the spectrum of the Floquet operator corresponding to (1) is purely absolutely continuous.