Decay estimates for bi-Schrödinger operators in dimension one.(English)Zbl 1505.35129
The authors consider the time decay estimates for the bi-Schrödinger operator\[H=\Delta^2+V(x),\quad H_0=\Delta^2,\]where \(V(x)\) is a real-valued function satisfying \(|V(x)| \lesssim (1+|x|)^{-\beta}\) for some \(\beta >0\).
The authors establish \(L^1-L^\infty\) estimates of \(H\) (with regularity terms) in dimension one. They show that the presence of zero resonance of \(H\) does not affect the time decay rate of \(e^{-itH}\) but requires higher decay rate of potential \(V\). As a consequence, Strichartz estimates are obtained for the fourth-order Schrödinger equations with potentials for initial data in \(L^2(R)\).
The authors establish \(L^1-L^\infty\) estimates of \(H\) (with regularity terms) in dimension one. They show that the presence of zero resonance of \(H\) does not affect the time decay rate of \(e^{-itH}\) but requires higher decay rate of potential \(V\). As a consequence, Strichartz estimates are obtained for the fourth-order Schrödinger equations with potentials for initial data in \(L^2(R)\).
Reviewer: Said El Manouni (Riyadh)
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