TY - JOUR
T1 - Global estimates for the Schrödinger equation
AU - Ben-Artzi, Matania
PY - 1992/8/1
Y1 - 1992/8/1
N2 - Let u = u(x, t) be a solution to the IVP for the Schrödinger equation iu1 = (-Δ + V(x))u ≡ Hu, u(x, 0) = u0(x) ε{lunate} PacL2(Rn) (Pac is the projection on the absolutely continuous subspace of H). Assume that for some ε > 0 the multiplication operator (1+|x|)1+iV(x):H1-ε(Rn) → L2(Rn) is bounded. Then u(x, t) = u1(x, t) + u2(x, t) where, for every s > 1 2, ∫ R ∫ Rn (1 + |x|2)-s|(1+H) 1 4u1(x,t)|2 dxdt ≤ C {norm of matrix}u0{norm of matrix}L2(Rn)2, and for every integer j, sup{norm of matrix}(I + H)ju2(·, t){norm of matrix}L2(Rn) ≤ Cj {norm of matrix} U0{norm of matrix}L2(Rn) tε{lunate}R.
AB - Let u = u(x, t) be a solution to the IVP for the Schrödinger equation iu1 = (-Δ + V(x))u ≡ Hu, u(x, 0) = u0(x) ε{lunate} PacL2(Rn) (Pac is the projection on the absolutely continuous subspace of H). Assume that for some ε > 0 the multiplication operator (1+|x|)1+iV(x):H1-ε(Rn) → L2(Rn) is bounded. Then u(x, t) = u1(x, t) + u2(x, t) where, for every s > 1 2, ∫ R ∫ Rn (1 + |x|2)-s|(1+H) 1 4u1(x,t)|2 dxdt ≤ C {norm of matrix}u0{norm of matrix}L2(Rn)2, and for every integer j, sup{norm of matrix}(I + H)ju2(·, t){norm of matrix}L2(Rn) ≤ Cj {norm of matrix} U0{norm of matrix}L2(Rn) tε{lunate}R.
UR - http://www.scopus.com/inward/record.url?scp=0012120135&partnerID=8YFLogxK
U2 - 10.1016/0022-1236(92)90113-W
DO - 10.1016/0022-1236(92)90113-W
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AN - SCOPUS:0012120135
SN - 0022-1236
VL - 107
SP - 362
EP - 368
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 2
ER -