论文标题
低能能处于低能
Hydrogen-like Schrödinger Operators at Low Energies
论文作者
论文摘要
考虑到渐近的欧几里德歧管上的$ x $ dimension $ x $至少两个，并假设潜力是有吸引力的库仑型。我们使用Vasy的第二个第二微胶质方法，“拉格朗日方法”，我们分析 - 一致，一直以下至$ e = 0 $ - 限制分辨率$ r（e \ pm i 0）= \ lim_ {ε\ to 0^+} r（e \ pm i的输出）。库仑电势会导致低能量分解的输出具有振荡渐近性，这些振荡渐近造成与吉拉尔穆（Guillarmou），Hassell，Sikora，Sikora和（最近）Hintz和Vasy在短期情况下观察到的各种渐近差异。具体而言，低能和大空间尺度的复合渐近造剂更细腻，并且分解输出一直平滑至$ e = 0 $。实际上，我们将构建一个$（0,1] _e \ times x $的紧凑型，由指定（相对复杂的）功能给出了分解输出，该功能将其振动为$ r \ to \ infty $ times times times timess lothomoyomenosient。作为循环，我们将获得完整的和兼容的渐近级别，以解决方案和零散的范围，以散布解决方案和零散的功能。
Consider a Schrödinger operator on an asymptotically Euclidean manifold $X$ of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy's second 2nd-microlocal approach, "the Lagrangian approach," we analyze -- uniformly, all the way down to $E=0$ -- the output of the limiting resolvent $R(E\pm i 0) = \lim_{ε\to 0^+} R(E\pm i ε)$. The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmou, Hassell, Sikora, and (more recently) Hintz and Vasy. Specifically, the compound asymptotics at low energy and large spatial scales are more delicate, and the resolvent output is smooth all the way down to $E=0$. In fact, we will construct a compactification of $(0,1]_E\times X$ on which the resolvent output is given by a specified (and relatively complicated) function that oscillates as $r\to\infty$ times something polyhomogeneous. As a corollary, we get complete and compatible asymptotic expansions for solutions to the scattering problem as functions of both position and energy, with a transitional regime.
