TY - JOUR
T1 - Smoothing effect for some Schrödinger equations
AU - Hayashi, Nakao
AU - Ozawa, Tohru
N1 - Funding Information:
* Present Address: Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376, Japan. ’ Partially supported by the Saneyoshi Foundation.
PY - 1989/8
Y1 - 1989/8
N2 - We study the Cauchy problem for the following Schrödinger equation in Rn(n∈N): i∂tu + 1 2Δu = V1u + (V2 * |u|2)u, (t,x)∈R×Rn, u(0) = φ, x∈Rn, (**) where V1 = V1(x) = λ1 |x|-γ1, V2 = V2(x) = ∑k = 23 λk |x|-γk, λk∈R (1 ≤ k ≤ 3), 0 < γ1 < min(2, n 2), 0 < γ2, γ3 < min(2, n). We prove the existence, uniqueness, and smoothing effect of global solutions of (**) with φ not necessarily in the Sobolev space H1(Rn).
AB - We study the Cauchy problem for the following Schrödinger equation in Rn(n∈N): i∂tu + 1 2Δu = V1u + (V2 * |u|2)u, (t,x)∈R×Rn, u(0) = φ, x∈Rn, (**) where V1 = V1(x) = λ1 |x|-γ1, V2 = V2(x) = ∑k = 23 λk |x|-γk, λk∈R (1 ≤ k ≤ 3), 0 < γ1 < min(2, n 2), 0 < γ2, γ3 < min(2, n). We prove the existence, uniqueness, and smoothing effect of global solutions of (**) with φ not necessarily in the Sobolev space H1(Rn).
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U2 - 10.1016/0022-1236(89)90039-6
DO - 10.1016/0022-1236(89)90039-6
M3 - Article
AN - SCOPUS:38249025921
SN - 0022-1236
VL - 85
SP - 307
EP - 348
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 2
ER -