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Existence of solutions for Schrödinger evolution equations. (English) Zbl 0638.35036
We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equation $$i \partial u/\partial t=(-1/2)\Delta u+V(t,x)u,\quad u(0)=u\sb 0.$$ We provide sufficient conditions on V(t,x) such that the equation generates a unique unitary propagator $U(t,s)$ and such that $$U(t,s)u\sb 0\in C\sp 1({\bbfR},L\sp 2)\cap C\sp 0({\bbfR},H\sp 2({\bbfR}\sp n))$$ for $u\sb 0\in H\sp 2({\bbfR}\sp n)$. The conditions are general enough to accomodate moving singularities of type $\vert x\vert\sp{-2+\epsilon}$ $(n\ge 4)$ or $\vert x\vert\sp{-n/2+\epsilon}\quad (n\le 3)$.
Reviewer: K.Yajima

35K15Second order parabolic equations, initial value problems
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35B65Smoothness and regularity of solutions of PDE
35B40Asymptotic behavior of solutions of PDE
Full Text: DOI
[1] Bellissard, J.: Stability and instability in quantum mechanics. In: Schrödinger operators. Graffi, S. (ed.) Lecture Notes in Mathematics, Vol.1049. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0581.35078
[2] Combescure, M.: A quantum particle in a quadrupole radio-frequency trap, Université de Paris-sud, preprint 1985 · Zbl 0613.46064
[3] Ginibre, J., Velo, G.: The global Cauchy problem for the non-linear Schrödinger equations revisited. Université de Paris-sud, preprint, 1984 · Zbl 0569.35070
[4] Goldstein, J.: Semigroups of linear operators and applications, Oxford: Oxford University Press 1985 · Zbl 0592.47034
[5] Howland, J.: Stationary scattering theory for time dependent Hamiltonians. Math. Ann.207, 315-335 (1974) · Zbl 0268.35071 · doi:10.1007/BF01351346
[6] Kato, T.: Linear evolution equations of ?hyperbolic type? I. J. Fac. Sci. Univ. Tokyo Sect. IA,17, 241-258 (1970): II, J. Math. Soc. Jpn.25, 648-666 (1973) · Zbl 0222.47011
[7] Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann.162, 258-279 (1966) · Zbl 0139.31203 · doi:10.1007/BF01360915
[8] Masuda, K.: Evolution equations, Tokyo: Kinokuniya-Shoten 1979 (in Japanese)
[9] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0516.47023
[10] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol.II. Fourier analysis and selfadjointness. New York: Academic Press 1975 · Zbl 0308.47002
[11] Strichartz, R.: Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J.44, 705-714 (1977) · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[12] Tanabe, H.: Evolution equations. Tokyo: Iwanami-Shoten 1975 (in Japanese)
[13] Tsutsumi, Y.: Global strong solutions for non-linear Schrödinger equations, Hiroshima University, preprint, 1985 · Zbl 0612.35104
[14] Yajima, K.: A multichannel scattering theory for some time-dependent Hamiltonians, charge transfer problem. Commun. Math. Phys.75, 153-178 (1980) · Zbl 0437.47008 · doi:10.1007/BF01222515
[15] Yajima, K.: Resonances for AC-Stark effect. Commun. Math. Phys.87, 331-352 (1982) · Zbl 0538.47010 · doi:10.1007/BF01206027
[16] Iorio, R. J., Marchesin, D.: On the Schrödinger equation with time-dependent electric fields. Proc. R. Soc. Edinb.96A, 117-134 (1984) · Zbl 0573.47005
[17] Wüller, U.: Existence of the time evolution for Schrödinger operators with time dependent singular potentials, preprint, Freie Universisät Berlin (1985)
[18] Kato, T.: On non-linear Schrödinger equations, preprint. University of California, Berkeley 1986