arXiv 2005.103232020-05math.APmath.CAmath.DGmath.SP

具有临界奇异位势的Schr dinger算子的Weyl公式

Weyl formulae for Schrödinger operators with critically singular potentials

Xiaoqi Huang, Christopher D. Sogge

备注：: A revised version which will appear in Communications in Partial Differential Equations

摘要：

我们得到的推广经典版本的Weyl公式涉及薛定谔运营商$H_V=-Δ_g+V(x)$紧无边界黎曼流形与临界奇异位$V$。特别地，我们通过在普遍情况下获得外尔公式中的误差项的$O(λ^{n-1})$界来扩展Avakumović，Levitan和Hörmander的经典结果，当我们仅仅假设$V$属于Kato类${\mathcal K}(M)$时，这是确保$H_V$本质上自伴且从下有界或具有有利的热核界的最小假设。在这种情况下，我们也可以得到Duistermaat-Guillemin定理的扩展，在测地线流上的一般条件下得到误差项的$o(λ^{n-1})$界，我们也可以在主曲率处处非正的假设下扩展Bérard定理，得到$O(λ^{n-1}/\log λ)$误差界。我们可以得到进一步的改进，环面，这是本质上最优的，如果我们加强的假设上的潜力$V\in L^p(M)\cap {\mathcal K}(M)$适当的指数$p=p_n$。

英文摘要：

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 $p=p_n$.

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