For a time periodic Hamiltonian \(H(t+\omega)=H(t)\), we gave a characterization of the bound and scattering states in terms of the spectral properties of the Floquet operator \(U(s+\omega,s)\) associated with H(t):
(1) \(H(t)=-\Delta +V(t)\), \(V(t+\omega)=V(t),\)
(2) \(i\partial u/\partial t=H(t)u\), \(u(s)=u\in {\mathcal H}=L^ 2(R^ n),\)
(3) \(u(t)=U(t,s)u.\)
Our result is an extension of D. Ruelle’s characterization of bound and scattering states for time-independent Hamiltonians [Nuovo Cimento, 59A, 655-662 (1969)]. Namely, \(u(t)=U(t,s)u\) is a bound (resp. scattering) state iff \(u\in {\mathcal H}_ p(U(s+\omega,s))\) (resp. \(u\in {\mathcal H}_ c(U(s+\omega,s))),\) the point (resp. continuous) spectral subspace of \(U(s+\omega,s)\). By that \(u(t)=U(t,s)u\) is a bound (resp. scattering) state, we meant that \[ \lim_{R\to \infty}\inf_{t}\| u(t)\|_{L^ 2(| x| \leq R)}=\| u\|_{{\mathcal H}}\quad resp.\quad \sup_{R>0}\lim_{T\to \pm \infty}T^{- 1}\int^{T}_{0}\| u(t)\|^ 2_{L^ 2(| x| \leq R)}dt=0. \] The time periodic potential V(t) is assumed to be a sum of a time periodic bounded operator \(V_ 1(t)\) on \({\mathcal H}\) and a potential \(V_ 2(t,x)\) which may have some singularities in x but decays as \(| x| \to \infty\) at least in a short-range fashion.
The totality of the scattering states in the above is easily seen to coincide with the space \({\mathcal H}^{\pm}_{scatt}(s)\) introduced in our paper [Duke Math. J. 49, 341-376 (1982; Zbl 0499.35087)] hence the completeness of the (modified) wave operators for the time periodic H(t) holds, whose proof in that paper was incomplete. It should also be noticed that such a result follows immediately, without referring to the Ruelle type characterization, from the relation \({\mathcal R}(W^{\pm}_ D(s))={\mathcal H}^{\pm}_{w,scatt}(s),\) which has been proved in our other paper in Duke Math. J. 50, 1005-1016 (1983).