垒顶共振的光谱投影、散射振幅的留数和薛定谔群展开
Spectral projection, residue of the scattering amplitude, and Schrodinger group expansion for barrier-top resonances
摘要：
研究了半经典薛定谔算符势垒顶共振的谱投影。首先，我们证明了一个预解估计复杂的能量接近这样的共振。使用该估计和共振态的显式表示，我们表明，光谱投影有一个半经典的展开在整数幂h，并计算其首项。我们使用这个结果来计算在这样的共振散射振幅的残差。最后，我们给出了薛定谔群在这些共振态下的大倍数展开式。
英文摘要：
We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrodinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrodinger group in terms of these resonances.
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