四阶非齐次薛定谔算子的KATO-JENSEN估计

KATO-JENSEN Estimates of Fourth Order Non-Homogeneous Schrödinger Operator

摘要: 本文研究带位势的四阶非齐次薛定谔算子$H={\Delta }^2-\Delta +V$ 生成的薛定谔群${\text{e}}^{-\text{i}tH}$ 在$R^5$ 中的Kato-Jensen估计，即在加权-$L^2$ 空间中建立${\text{e}}^{-\text{i}tH}$ 关于时间的衰减估计。通过建立相应的谱测度估计，对算子唯一的能量阈值0分为正则点和特征值进行讨论。当零能量阈值为H的正则点时，时间衰减指数为−5/2。当零能量阈值为H的特征值时，时间衰减指数则为−1/2。

Abstract: In this paper we study the Kato-Jensen estimates of the Schrödinger group ${\text{e}}^{-\text{i}tH}$ in $R^5$ which is generated by the non-homogeneous fourth-order Schrödinger operator with potential $H={\Delta }^2-\Delta +V$ . The Kato-Jensen estimate means time decay estimate for ${\text{e}}^{-\text{i}tH}$ in the weighted-$L^2$ space. By deriving the corresponding spectral measure estimates, we classify the unique energy threshold 0 into regular points and eigenvalues. When the zero energy threshold is a regular point of H, the time decay exponent is −5/2. Conversely, when the zero energy threshold is an eigenvalue of H the time decay exponent is −1/2.