Existence of solutions for Schrödinger evolution equations. (English) Zbl 0638.35036

Yajima, Kenji

Commun. Math. Phys. 110, 415-426 (1987).

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equation \[ i \partial u/\partial t=(-1/2)\Delta u+V(t,x)u,\quad u(0)=u_ 0. \] We provide sufficient conditions on V(t,x) such that the equation generates a unique unitary propagator \(U(t,s)\) and such that \[ U(t,s)u_ 0\in C^ 1({\mathbb{R}},L^ 2)\cap C^ 0({\mathbb{R}},H^ 2({\mathbb{R}}^ n)) \] for \(u_ 0\in H^ 2({\mathbb{R}}^ n)\). The conditions are general enough to accomodate moving singularities of type \(| x|^{-2+\epsilon}\) \((n\geq 4)\) or \(| x|^{-n/2+\epsilon}\quad (n\leq 3)\).

Reviewer: K.Yajima

MSC:

35K15	Initial value problems for second-order parabolic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35B65	Smoothness and regularity of solutions to PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

existence; uniqueness; regularity; initial value problem; time dependent Schrödinger equation; unitary propagator; moving singularities

References:

[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) · 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