AI 中文总结
研究史瓦西黑洞阈值能量附近准正则模数量的魏尔定律。通过新伪微分算子演算及复标度方法，给出预解式估计，应用于雷杰 - 惠勒势，结合渐近描述，得出QNM数量增长规律，并研究截断预解式的影响。
AI 中文摘要
我们证明了史瓦西黑洞在实轴下方扇形区域内准正则模（QNM）数量的魏尔定律。这需要引入一种新的伪微分算子演算，用于研究阈值能量附近的半经典谱问题。该演算中的椭圆理论可与复标度方法相结合，为在无穷远处表现为具有排斥性平方反比势的半经典薛定谔算子的算子给出零能量附近的一致预解式估计。应用于雷杰 - 惠勒势时，我们的方法表明，从半径随角动量线性增长的圆盘上不存在高角动量QNM。结合希特里克和兹沃尔斯基最近得到的史瓦西QNM的渐近描述，这表明在实轴下方且模由\(\lambda\)界定的小扇形区域内的QNM数量增长为\(C\lambda^3\)。我们还研究了在远离事件视界处截断史瓦西预解式的影响，并表明这种截断不会导致任何极点抵消。
英文摘要
We prove a Weyl law for the number of quasinormal modes (QNM) of a Schwarzschild black hole contained in a sector below the real axis. This requires introducing a new pseudodifferential operator calculus tailored to the study of semiclassical spectral problems near threshold energies. Elliptic theory in this calculus can be combined with the method of complex scaling to give uniform resolvent estimates near zero energy for operators that behave at infinity like a semiclassical Schrödinger operator with a repulsive inverse-square potential. Applied to the Regge-Wheeler potential, our methods imply the absence of high angular momentum QNM from a disc whose radius grows linearly with the angular momentum. Together with the asymptotic description of Schwarzschild QNM recently obtained by Hitrik and Zworski, this shows that the number of QNM contained in a small sector below the real axis and with modulus bounded by $λ$ grows as $Cλ^3$. We also study the effect of cutting off the Schwarzschild resolvent away from the event horizon and show that such a cutoff does not lead to any pole cancellations.
Comments38 pages, 1 figure
