Resonances for Obstacles in Hyperbolic Space
Resonances for Obstacles in Hyperbolic Space
复制标题
双曲空间中障碍物的共振
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
M. Zworski
中科院分区：
文献类型：
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作者：
P. Hintz;M. Zworski
我们考虑了双曲空间中星形障碍物的散射，并证明了共振满足一个泛界$${ {
m Im} lambda leq - frac12 }$$ Imλ≤-12，这在维数2上是最优的。在奇维，我们还证明了$${ {
m Im} lambda leq - frac{mu}{
ho} }$$ Imλ≤-μρ对于一个普遍常数$${mu}$$ μ，其中$${
ho }$$ ρ是包含障碍物的球的半径；这给了小障碍一个改进。在3维及更高的维度上，证明遵循Morawetz的经典向量场方法，而在2维中，我们通过处理来自广义相对论的空间来获得边界。我们还表明，在奇维中，小障碍物的共振在适当的意义上接近于欧几里得共振。
We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound $${ {
m Im} lambda leq - frac12 }$$Imλ≤-12, which is optimal in dimension 2. In odd dimensions we also show that $${ {
m Im} lambda leq - frac{mu}{
ho} }$$Imλ≤-μρ for a universal constant $${mu}$$μ, where $${
ho }$$ρ is the radius of a ball containing the obstacle; this gives an improvement for small obstacles. In dimensions 3 and higher the proofs follow the classical vector field approach of Morawetz, while in dimension 2 we obtain our bound by working with spaces coming from general relativity. We also show that in odd dimensions resonances of small obstacles are close, in a suitable sense, to Euclidean resonances.