论文标题
具有奇特电势的Schrödinger操作员的Weyl公式
Weyl formulae for Schrödinger operators with critically singular potentials
论文作者
论文摘要
我们获得了涉及Schrödinger操作员的Weyl公式的经典版本的概括，$ H_V =-Δ_G+V（X）$在紧凑的无边界Riemannian歧管上具有重要的单数电位$ V $。特别是，我们通过获得$ o（λ^{n-1}）$在普遍情况下的错误术语中获得$ o（λ^{n-1}）$的经典结果来扩展$ o（λ^{n-1}）$的界限，当时我们只假设$ v $仅假设$ v $属于$ {\ nathcal k}（\ nathcal k}（\ nathcal k}（\ mathcal k}（$）$）从下方界定或具有有利的热核边界。在这种情况下，我们还可以获取duistermaat-guillemin定理的扩展，该定理产生了$ o（λ^{n-1}）$在通用条件下的误差期限的界限，并且我们也可以在各个范围内的$ o（λ^{n-1}/\ logution cunteruly cultive cultive culive culive culive cul butivebérard的bérardtrud bunters $ o（λ^{n-1}/\λ）的范围不在如果我们加强了对$ v \ $ v \ in l^p（m）\ cap {\ mathcal k}（m）$的$ v \的假设，那么我们可以获得基本上是最佳的Tori的进一步改进。
We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators $H_V=-Δ_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials $V$. In particular, we extend the classical results of Avakumović , Levitan and Hörmander by obtaining $O(λ^{n-1})$ bounds for the error term in the Weyl formula in the universal case when we merely assume that $V$ belongs to the Kato class, ${\mathcal K}(M)$, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. In this case, we can also obtain extensions of the Duistermaat-Guillemin theorem yielding $o(λ^{n-1})$ bounds for the error term under generic conditions on the geodesic flow, and we can also extend Bérard's theorem yielding $O(λ^{n-1}/\log λ)$ error bounds under the assumption that the principal curvatures are non-positive everywhere. We can obtain further improvements for tori, which are essentially optimal, if we strengthen the assumption on the potential to $V\in L^p(M)\cap {\mathcal K}(M)$ for appropriate exponents $p=p_n$.
