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On the Schrödinger equation with time-dependent electric fields. (English) Zbl 0573.47005

The authors first establish the existence and uniqueness of the solution of the initial value problem for the Schrödinger equation \[ i(\partial /\partial t)\psi (x,t)=(-\Delta +g(t)x_ 1+q(x))\psi (x,t),\quad x=(x_ 1,x_ 2,x_ 3)\in {\mathbb{R}}^ 3, \] with a time-dependent electric field g(t)(1,0,0), \(g\in C^ 1({\mathbb{R}})\), and a potential q(x) satisfying a certain condition including the Coulomb case. It is done by applying T. Kato’s theorem on linear evolution equations of ”hyperbolic” type [J. Fac. Sci. Univ. Tokyo, Sect. IA 17, 241-258 (1970; Zbl 0222.47011); J. Math. Soc. Japan 25, 648-666 (1973; Zbl 0262.34048)].
Next the scattering theory is studied for g(t) periodic with period \(T>0\) and mean value zero, and q(x) short-range and well-behaved in the \(x_ 1\) direction. They prove the existence and completeness of the wave operators to characterize the scattering and bound states. It is based on J. S. Howland’s theory of considering a space-time hamiltonian containing the term -i\(\partial /\partial t\) on the enlarged Hilbert space \(L^ 2([0,T])\otimes L^ 2({\mathbb{R}}^ 3)\) [Math. Ann. 207, 315- 335 (1974; Zbl 0261.35067)]. As mentioned in the paper a different approach to the same problem is made when \(g(t)=\cos (2\pi t/T)=\cos \omega t\), \(T=2\pi /\omega\), by H. Kitada and K. Yajima [Duke Math. J. 49, 341-376 (1982; Zbl 0499.35087); ibid. 50, 1005-1016 (1983; Zbl 0554.35096)].
Reviewer: T.Ichinose

MSC:

47A40 Scattering theory of linear operators
35P25 Scattering theory for PDEs
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References:

[1] DOI: 10.2969/jmsj/02940729 · Zbl 0356.47010 · doi:10.2969/jmsj/02940729
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