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Volume 41, Issue 1
Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three

Ping Li, Zijun Wan, Hua Wang & Xiaohua Yao

Ann. Appl. Math., 41 (2025), pp. 1-41.

Published online: 2025-04

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  • Abstract

This paper is concerned with the time decay estimates of the fourth order Schrödinger operator $H = ∆^2+V (x)$ in dimension three, where $V (x)$ is a real valued decaying potential. Assume that zero is a regular point or the first kind resonance of $H,$ and $H$ has no positive eigenvalues, we established the following time optimal decay estimates of $e^{−it H}$ with a regular term $H^{α/4}:$

image.png

When zero is the second or third kind resonance of $H,$ their decay will be significantly changed. We remark that such improved time decay estimates with the extra regular term $H^{α/4}$ will be interesting in the well-posedness and scattering of nonlinear fourth order Schrödinger equations with potentials.

  • AMS Subject Headings

58J50, 42B15, 35P15, 42B20, 47F05

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COPYRIGHT: © Global Science Press

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@Article{AAM-41-1, author = {Li , PingWan , ZijunWang , Hua and Yao , Xiaohua}, title = {Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three}, journal = {Annals of Applied Mathematics}, year = {2025}, volume = {41}, number = {1}, pages = {1--41}, abstract = {

This paper is concerned with the time decay estimates of the fourth order Schrödinger operator $H = ∆^2+V (x)$ in dimension three, where $V (x)$ is a real valued decaying potential. Assume that zero is a regular point or the first kind resonance of $H,$ and $H$ has no positive eigenvalues, we established the following time optimal decay estimates of $e^{−it H}$ with a regular term $H^{α/4}:$

image.png

When zero is the second or third kind resonance of $H,$ their decay will be significantly changed. We remark that such improved time decay estimates with the extra regular term $H^{α/4}$ will be interesting in the well-posedness and scattering of nonlinear fourth order Schrödinger equations with potentials.

}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2024-0018}, url = {http://global-sci.org/intro/article_detail/aam/23961.html} }
TY - JOUR T1 - Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three AU - Li , Ping AU - Wan , Zijun AU - Wang , Hua AU - Yao , Xiaohua JO - Annals of Applied Mathematics VL - 1 SP - 1 EP - 41 PY - 2025 DA - 2025/04 SN - 41 DO - http://doi.org/10.4208/aam.OA-2024-0018 UR - https://global-sci.org/intro/article_detail/aam/23961.html KW - Fourth order Schrödinger equation, asymptotic expansions, $L^1−L^∞$ decay estimate, resonances. AB -

This paper is concerned with the time decay estimates of the fourth order Schrödinger operator $H = ∆^2+V (x)$ in dimension three, where $V (x)$ is a real valued decaying potential. Assume that zero is a regular point or the first kind resonance of $H,$ and $H$ has no positive eigenvalues, we established the following time optimal decay estimates of $e^{−it H}$ with a regular term $H^{α/4}:$

image.png

When zero is the second or third kind resonance of $H,$ their decay will be significantly changed. We remark that such improved time decay estimates with the extra regular term $H^{α/4}$ will be interesting in the well-posedness and scattering of nonlinear fourth order Schrödinger equations with potentials.

Li , PingWan , ZijunWang , Hua and Yao , Xiaohua. (2025). Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three. Annals of Applied Mathematics. 41 (1). 1-41. doi:10.4208/aam.OA-2024-0018
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