Geometric scattering theory for long-range potentials and metrics
Geometric scattering theory for long-range potentials and metrics
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长程势和度量的几何散射理论
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发表时间：
1998
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通讯作者：
A. Vasy
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作者：
A. Vasy
设X是一个有边界的紧致流形，n = dimX，x是一个边界定义函数，即x ∈ C（X），x ≥ 0，<$X = {p ∈ X：x（p）= 0}，且dx在<$X处不为零.我们考虑X上的以下一类渐近平坦的完备度量，它们为Melrose在[13]中首次讨论的欧几里得散射理论的自然推广提供了背景。在X内部的黎曼度量g称为长程散射度量，如果它可以在X上的边界定义函数x，a−1 ∈ xC（X），以及X上的某个光滑对称2-余张量h限制在X上的度量h的情况下，在X附近被化为g = a 2 x4 + h x2的形式。此外，在[13]之后，如果我们可以取a = 1，则我们说g是（短程）散射度量。这种设置的一个特殊例子是欧几里得空间R到半球（即球）X = S+的径向紧化，通过球极投影SP的（非标准）版本，参见[13]，以及标准欧几里得度量的相应提升。更一般地说，在<$X附近，我们可以把X写成[0，）x × <$X。引入r = x，从而将<$X移动到“无穷大”，这个区域可以被认为是（，∞）r × <$X。然后，如果我们重新引入x = r，则对于大的r，形式为dr 2 + rh的度量，h是在λ X上的度量，成为（短程）散射度量，即当配备这样的度量时，我们可以将这个区域视为圆锥的“大端”（尖端将是r = 0）。除了在研究任意流形上的散射度量的内在几何兴趣之外，理解这些可以通过去除特殊的对称性来澄清欧几里得散射理论。它也可以为更详细地描述多体散射等复杂问题提供基础（见[3，18]，特别是[16，17]）。令A为g的（正）拉普拉斯算子，令H = A +V，其中V为二阶微分算子，使得H是自伴的，满足非退化（椭圆性）条件，并且H在X处的行为在某种自然意义上受A的支配。如果V是一个实值函数的乘积，这些要求相当于V ∈ xC（X）的陈述，即V在边界<$X处为零;一般（和精确）的设置将在下一节讨论。我们注意到V ∈ xC（S+）意味着V是从R拉回的一个−1阶经典（多齐次）符号，因此R上的库仑型势（在原点没有奇点）适合这个框架。这样的扰动V在[13]的意义上是“长程”的;“短程”V的一个例子是V ∈ xC（X）。
LetX be a compact manifold with boundary, n = dimX , and let x be a boundary defining function, i.e. x ∈ C(X), x ≥ 0, ∂X = {p ∈ X : x(p) = 0}, and dx is not zero at ∂X . We consider the following class of asymptotically flat, complete metrics onX which provide the background for a natural generalization of Euclidian scattering theory, first discussed by Melrose in [13]. A Riemannian metric g in the interior of X is called a long-range scattering metric if it can be brought to the form g = a 2 x4 + h x2 near ∂X for some choice of a boundary defining function x, a−1 ∈ xC(X), and for some smooth symmetric 2-cotensor h on X which restricts to a metric h on ∂X . Also, following [13], we say that g is a (short-range) scattering metric if we can take a = 1 above. A particular example of this setup is the radial compactification of Euclidian space R to a hemisphere (i.e. a ball) X = S+ by a (non-standard) version of the stereographic projection SP, see [13], and the corresponding lifting of the standard Euclidian metric. More generally, near ∂X we can write X as [0, )x × ∂X . Introducing r = x and thereby moving ∂X to ‘infinity’, this region can be regarded as ( ,∞)r × ∂X . Then metrics of the form dr 2 + rh for large r, h a metric on ∂X , become (short-range) scattering metrics if we reintroduce x = r, i.e. we can regard this region, when equipped with such a metric, as the ‘large end’ of a cone (the tip would have been r = 0). Apart from the intrinsic geometric interest in the study of scattering metrics on arbitrary manifolds with boundary, understanding these can clarify Euclidian scattering theory by removing the special symmetries. It can also provide the foundations for a more detailed description of such complex subjects as many-body scattering (see [3, 18] and especially [16, 17]). Let ∆ be the (positive) Laplacian of g, and let H = ∆+V , where V is a secondorder differential operator, be such that H is self-adjoint, satisfies non-degeneracy (ellipticity) conditions and that the behavior of H at ∂X is dominated by that of ∆ in a certain natural sense. If V is multiplication by a real-valued function, these requirements amount to the statement that V ∈ xC(X), i.e. that V vanishes at the boundary ∂X ; the general (and precise) setup is discussed in the following section. We remark that V ∈ xC(S+) means that V is the pull back of a classical (polyhomogeneous) symbol of order −1 from R, so Coulomb-type potentials on R (without the singularity at the origin) fit into this framework. Such a perturbation V is ‘long-range’ in the sense of [13]; an example of a ‘short-range’ V is V ∈ xC(X).