Global estimates for the Schrödinger equation

Matania Ben-Artzi

doi:10.1016/0022-1236(92)90113-W

Global estimates for the Schrödinger equation

Matania Ben-Artzi^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

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} P_acL²(Rⁿ) (P_ac is the projection on the absolutely continuous subspace of H). Assume that for some ε > 0 the multiplication operator (1+|x|)¹⁺ⁱV(x):H^1-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 4}u₁(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) ≤ C_j {norm of matrix} U₀{norm of matrix}_L²(Rⁿ) tε{lunate}R.

Original language	English
Pages (from-to)	362-368
Number of pages	7
Journal	Journal of Functional Analysis
Volume	107
Issue number	2
DOIs	https://doi.org/10.1016/0022-1236(92)90113-W
State	Published - 1 Aug 1992

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@article{27f6bdd70b6145f0a1be1b97488236e9,

title = "Global estimates for the Schr{\"o}dinger equation",

abstract = "Let u = u(x, t) be a solution to the IVP for the Schr{\"o}dinger 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.",

author = "Matania Ben-Artzi",

year = "1992",

month = aug,

day = "1",

doi = "10.1016/0022-1236(92)90113-W",

language = "אנגלית",

volume = "107",

pages = "362--368",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "2",

}

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 - https://www.scopus.com/pages/publications/0012120135

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 -

Global estimates for the Schrödinger equation

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