## Bound states and scattering states for time periodic Hamiltonians.(English)Zbl 0544.35073

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).

### MSC:

 35P25 Scattering theory for PDEs 47A40 Scattering theory of linear operators 35Q99 Partial differential equations of mathematical physics and other areas of application 35J10 Schrödinger operator, Schrödinger equation

Zbl 0499.35087
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### References:

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