|
| Time Decay Estimates for Fourth-OrderSchr¨odinger Operators in Dimension Three |
| Ping Li,Zijun Wan,Hua Wang,Xiaohua Yao |
| (School of Informations and Mathematics, Yangtze University, Jingzhou,
Hubei 434000, China;School of Mathematics and Statistics, Central China Normal University,
Wuhan, Hubei 430079, China;Department of Mathematics and Hubei Province Key Laboratory of
Mathematical Physics, Central China Normal University, Wuhan,
Hubei 430079, China;Department of Mathematics and Key Laboratory of Nonlinear Analysis
and Applications (Ministry of Education), Central China Normal
University, Wuhan, Hubei 430079, China) |
| DOI： |
| Abstract: |
| This paper is concerned with the time decay estimates of the fourth
order Schr¨odinger 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
?itH with a regular term Hα/4
:
k
Hα/4
e
?itHPac(H)k L1?L∞ . |t|
?
3+α
4 , 0≤α≤3.
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 Hα/4 will be interesting in the well-posedness and
scattering of nonlinear fourth order Schr¨odinger equations with potentials. |
| Key words: Fourth order Schr¨odinger equation, asymptotic expansions, L
1−L∞ decay
estimate, resonances. |