Semiclassical resolvent estimates for bounded potentials
Semiclassical resolvent estimates for bounded potentials
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有界势的半经典解析估计
DOI：
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发表时间：
2018
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通讯作者：
Martin Vogel
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作者：
F. Klopp;Martin Vogel
研究了$mathbb{R}^d$上具有紧支持有界势$V$的半经典Schr{o}dinger算子的截止解。我们证明了对于紧化区间$mathbb{R}_+$上的实能量$lambda^2$，以及对于支持在势$V$附近的球上的任意光滑截止函数$chi$，对于某常数$C>0$，可以得到egin{方程*}| chi (-h^2Delta + V-lambda^2)^{-1} chi |_{L^2 o H^1} leq C, mathm {e}^{Ch^{-4/3}log frac{1}{H}}。这个界特别显示了共振$lambda$虚部的上界，定义为解析$(- H^ 2Delta + V-lambda^2)^{-1}$的亚纯连续的极点，作为一个算子$L^2_{mathm {comp}} o H^2_{mathm {loc}}$：在离$0$的紧区间内，任何具有实部的共振$lambda$的虚部最多为egin{equation*} mathm {Im} lambda leq - C^{-1}， mathm {e}^{Ch^{-4/3}log frac{1}{H}}。这与Landis的一个猜想有关：在我们的证明中，主Carleman估计也提供了$L^2$解$u$到$-Delta u = Vu$的衰减率的下界
We study the cut-off resolvent of semiclassical Schr{"o}dinger operators on $mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $lambda^2$ in a compact interval in $mathbb{R}_+$ and for any smooth cut-off function $chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has egin{equation*} | chi (-h^2Delta + V-lambda^2)^{-1} chi |_{L^2 o H^1} leq C ,mathrm{e}^{Ch^{-4/3}log frac{1}{h} }. end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $lambda$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2Delta + V-lambda^2)^{-1}$ as an operator $L^2_{mathrm{comp}} o H^2_{mathrm{loc}}$: any resonance $lambda$ with real part in a compact interval away from $0$ has imaginary part at most egin{equation*} mathrm{Im} lambda leq - C^{-1} ,mathrm{e}^{Ch^{-4/3}log frac{1}{h} }. end{equation*} This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of $L^2$ solutions $u$ to $-Delta u = Vu$ with $0
otequiv Vin L^{infty}(mathbb{R}^d)$. We show that there exist a constant $M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has egin{equation*} int_{B(0,R+1)ackslash overline{B(0,R)}}|u(x)|^2 dx geq M^{-1}R^{-4/3} mathrm{e}^{-M |V|_{infty}^{2/3} R^{4/3}}|u|^2_2. end{equation*}