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On the unitarity of the wave operators for AC-Stark Hamiltonians

Published online by Cambridge University Press:  14 November 2011

Min-Jei Huang
Min-Jei Huang
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043
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Extract

We study a class of AC-Stark Hamiltonians H1(t) = H0(t) + V, where H0(t)= −Δ + E · x cos ωt. For a class of repulsive potentials, we show that the wave operators exist and are unitary, provided that |E|/ω2 is small. If |E|/ω2 is sufficiently large, the result remains true as long as V is sufficiently small.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 3 , 1995 , pp. 617 - 624
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.. Schrödinger Operators with Application to Quantum Mechanics and Global Geometry (Berlin: Springer-Verlag, 1987).Google Scholar
2Faris, W. G. and Lavine, R. B.. Commutators and self-adjointness of Hamiltonian operators. Comm. Math. Phys. 35 (1974), 39–48.CrossRefGoogle Scholar
3Howland, J. S.. Two problems with time-dependent Hamiltonians. In Mathematical Methods and Applications of Scattering Theory, ed DeSanto, J. A., Saenz, A. W. and Zachary, W. W., Lecture Notes in Physics 130 (Berlin: Springer, 1980).CrossRefGoogle Scholar
4Huang, M.-J.. On the absolutely continuous subspaces of Floquet operators. Proc. Roy. Soc. Edinburgh Sect. A 124(1994), 703–12.CrossRefGoogle Scholar
5Huang, M.-J. and Lavine, R. B.. Boundedness of kinetic energy for time-dependent Hamiltonians. Indiana Univ. Math. J. 38 (1989), 189–210.CrossRefGoogle Scholar
6Iório, R. J. Jr. and Marchesin, D.. On the Schrödinger equation with time-dependent electric fields. Proc. Roy. Soc. Edinburgh Sect. A 96 (1984), 117–34.CrossRefGoogle Scholar
7Kato, T.. Linear evolution equations of hyperbolic type, II. J. Math. Soc. Japan 25 (1973), 648–66.CrossRefGoogle Scholar
8Kato, T. and Yajima, K.. Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1 (1989), 481–96.CrossRefGoogle Scholar
9Kitada, H. and Yajima, K.. A scattering theory for time-dependent long-range potentials. Duke Math. J. 49 (1982), 341–76.CrossRefGoogle Scholar
10Yajima, K.. Scattering theory for Schrödinger equations with potentials periodic in time. J. Math. Soc. Japan 29 (1977), 729–43.CrossRefGoogle Scholar