ar X iv : 0 90 8 . 32 11 v 1 [ m at h . A P ] 2 1 A ug 2 00 9 WEAK ASYMPTOTICS FOR SCHRÖDINGER EVOLUTION

In this short note, we apply technique developed in [2] to study the long-time evolution for Schrödinger equation with slowly decaying potential. Consider H = −∂ xx + q, x > 0 with Dirichlet boundary condition at zero. If q = 0, we denote the operator by H0. Through the paper, the potentials is real valued and satisfies the following condition q(x) ln(|x|+ 2) ∈ L(R) (1) We will use the asymptotics of generalized eigenfunctions obtained in [2] to prove existence of modified wave operators. Unfortunately, the limits will be understood in some averaged sense only. In the meantime, the methods are rather robust and can be used for other dispersive equations. For q ∈ L(R), 1 ≤ p < 2 the existence of modified wave operators was proved in [1]. We will start with some definitions. Assume that f(x) ∈ L(R) and take its odd continuation to R. Call it fo(x). Then e 2 fo ∼ κ e /(4t) √ t f̂o(x/(2t)) in L (R), t→ ∞ (2) where κ = − 1 (1 + i) √ 2π , f̂o(ω) = ∫ fo(x)e dx so f̂o denotes the inverse Fourier transform. (In this paper, f ∼ g as t → ∞ if ‖f − g‖ → 0 as t → ∞ in the specified metric). The asymptotics (2) is easy to check if f̂o is infinitely smooth and compactly supported away from zero. The general L case then follows upon making the simple observation that the l.h.s. and the r.h.s. are unitary in f and then using the approximation argument. Then, by symmetry, e−iH0tf ∼ κ ix/(4t) √ t f̂o(x/(2t))χx>0, in L (R), t→ +∞ We will need to modify the free evolution. The modification will be made in the physical space as follows U(t)f = κ e /(4t) √ t f̂o(x/(2t)) exp ( −i t x ∫ x 0 q(s)ds )