光谱底部的正向散射体
Positive commutators at the bottom of the spectrum
摘要：
Bony和Häfner最近在渐近欧氏空间的低能极限中得到了拉普拉斯算子的正交换子估计;如果度量是非捕获的，这些估计可以用来证明局部能量衰减估计。我们简化了Bony-Häfner估计的证明，并通过应用尖锐的Poincaré不等式将其推广到散射流形（即具有大圆锥端的流形）的设置。我们的主要结果是正交换子估计$$ χ_I(H^2Δ_g)\frac{i}{2}[H^2Δ_g,A]χ_I(H^2Δ_g) \geq Cχ_I(H^2Δ_g)^2, $$其中$H\uparrow \infty$是\emph{large}参数，$I$是$(0,\infty),$中的紧区间，$χ_I$是其指示函数，$A$是紧集外的微分算子，在无穷远处等于$(1/2)(r D_r +(r D_r)^*)$.拉普拉斯算子也可以通过增加一个衰减足够快的正势来修改--同样的估计也适用于所得的薛定谔算子。
英文摘要：
Bony and Häfner have recently obtained positive commutator estimates on the Laplacian in the low-energy limit on asymptotically Euclidean spaces; these estimates can be used to prove local energy decay estimates if the metric is non-trapping. We simplify the proof of the estimates of Bony-Häfner and generalize them to the setting of scattering manifolds (i.e. manifolds with large conic ends), by applying a sharp Poincaré inequality. Our main result is the positive commutator estimate $$ χ_I(H^2Δ_g)\frac{i}{2}[H^2Δ_g,A]χ_I(H^2Δ_g) \geq Cχ_I(H^2Δ_g)^2, $$ where $H\uparrow \infty$ is a \emph{large} parameter, $I$ is a compact interval in $(0,\infty),$ and $χ_I$ its indicator function, and where $A$ is a differential operator supported outside a compact set and equal to $(1/2)(r D_r +(r D_r)^*)$ near infinity. The Laplacian can also be modified by the addition of a positive potential of sufficiently rapid decay--the same estimate then holds for the resulting Schrödinger operator.
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