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Decay estimates for higher-order elliptic operators. (English) Zbl 1440.35054

The paper under review is devoted to the study of time decay estimates for higher-order Schrödinger-type operators in \(\mathbb{R}^{n}\): \[H=(-\Delta)^{m}+V(x),\] where \(n>2m\), \(m\in\mathbf{N}\) and \(V(x)\) is a real-valued function satisfying \(|V(x)|\lesssim (1+|x|)^{-\beta}\) for some \(\beta>0\). Let \(s, s'\in\mathbb{R}\) and \(B(s, s')\) denote the space of bounded operators from \(L^{2}_{s}(\mathbb{R}^n)\) to \(L^{2}_{s'}(\mathbb{R}^n)\), where \(L^{2}_{s}(\mathbb{R}^n)\) is the weighted \(L^2\) space: \(L^{2}_{s}(\mathbb{R}^n)=\Big\{f: (1+|\cdot|)^{s}f\in L^{2}(\mathbb{R}^n)\Big\}.\)
At first, the authors derive asymptotic expansions of the resolvent \(R_V(z)=\big((-\Delta)^m+V-z\big)^{-1}\) at zero energy and decay rate of \(R_V(z)\) in \(B(s,-s')\) as \(z\) goes to infinity with some suitable \(s, s' > 0\). The limiting absorption principle for \(R_V(z)\) is also proved.
Let \(P_{ac}\) be the projection onto the absolutely continuous spectrum space of \(H\). The main results of the paper are Kato-Jensen type estimates, i.e., estimates of the following type. \[ \left\| e^{itH}P_{ac}(H)\right\|_{B(s,-s')} \leq C (1+|t|)^\alpha, t\in \mathbb{R}. \] These estimates are proved in the presence of zero resonance or zero eigenvalue.
Another important result is the following local decay estimate: Let \(|V(x)|\lesssim (1+|x|)^{-\beta}\) for some \(\beta>n\). Assume zero is a regular point of \(H\) and \(H\) has no positive embedded eigenvalue. Then for \(\sigma>n/2\) \[\Big\| (1+|x|^2)^{-\sigma/2} e^{-itH}P_{ac}(H)\phi\Big\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}^{n+1})}\lesssim\|\phi\|_{L^{2}(\mathbb{R}^{n})}.\]
With the help of these results, endpoint Strichartz estimates and \(L^1\cap L^2- L^{\infty}+L^2\)-decay estimates for the operator \(e^{itH}P_{ac}(H)\) are established.
Finally, using the virial identity a criterion on the absence of positive embedded eigenvalues is given for \(h(D) + V\), where \(h \geq 0\) is a homogeneous real-valued function and \(V(x)\) is repulsive, that is, \(V(\gamma x)\leq V(x)\) for all \(\gamma>1\) and \(x\in\mathbb{R}^{n}\).

MSC:

35J10 Schrödinger operator, Schrödinger equation
35J30 Higher-order elliptic equations
35P05 General topics in linear spectral theory for PDEs
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