Abstract

We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators $H_V=-{\Delta }_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials V. In particular, we extend the classical results of Avakumović [1956 Avakumović, V. G. (1956). Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65(1):327–344. DOI: https://doi.org/10.1007/BF01473886.[Crossref] , [Google Scholar]], Levitan [1952 Levitan, B. M. (1952). On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order. Izvestiya Akad. Nauk SSSR. Ser. Mat. 16:325–352. [Google Scholar]] and Hörmander [1968 Hörmander, L. (1968). The spectral function of an elliptic operator. Acta Math. 121(0):193–218. DOI: https://doi.org/10.1007/BF02391913.[Crossref] , [Google Scholar]] by obtaining $O({\lambda }^{n-1})$ bounds for the error term in the Weyl formula in the universal case when we assume that $V\in L^1(M)$ with the negative part $V^{-}=\max \{0,-V\}$ 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 [1975 Duistermaat, J. J., Guillemin, V. W. (1975). The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1):39–79. DOI: https://doi.org/10.1007/BF01405172.[Crossref], [Web of Science ®] , [Google Scholar]] theorem yielding $o({\lambda }^{n-1})$ bounds for the error term under generic conditions on the geodesic flow, and we can also extend Bérard’s (1977 Bérard, P. H. (1977). On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3):249–276. DOI: https://doi.org/10.1007/BF02028444.[Crossref], [Web of Science ®] , [Google Scholar]) theorem yielding $O({\lambda }^{n-1}/\hspace{0.17em}\text{log}\hspace{0.17em}\lambda )$ 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 $V\in L^p(M)$ and $V^{-}\in \mathcal{K}(M)$ for appropriate exponents p = p_n.

摘要

我们获得了涉及薛定谔算子的 Weyl 公式的经典版本的推广 $H_{伏}=-{\Delta }_G+伏(X)$在具有临界奇异势V 的紧凑无边界黎曼流形上。特别是，我们扩展了 Avakumović [ 1956]的经典结果 阿瓦库莫维奇，VG（1956 年）。über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten。数学。Z. 65(1): 327 – 344。DOI：https://doi.org/10.1007/BF01473886。[Crossref]  , [Google Scholar] ], Levitan [1952] 列维坦，BM（1952 年）。关于二阶自伴随微分方程谱函数的渐近行为。消息报。诺克 SSSR。爵士。垫子。16: 325 – 352。 [Google Scholar]和 Hörmander [1968] 赫尔曼德，L.（1968 年）。椭圆算子的谱函数。数学学报。121(0): 193 – 218。DOI：https://doi.org/10.1007/BF02391913。[Crossref]  , [Google Scholar] ] 通过获得$哦({\lambda }^{n-1})$ 在我们假设的普遍情况下，Weyl 公式中误差项的界限 $伏\in {升}^1(米)$ 与消极的部分 ${伏}^{-}=\mathrm{最大限度}\{0,-伏\}$ 属于加藤班， $\mathcal{钾}(米),$这是确保H _V 本质上是自伴的并从下方有界或具有有利的热核边界的最小假设。在这种情况下，我们还可以获得 Duistermaat-Guillemin [ 1975 Duistermaat, JJ , Guillemin, VW ( 1975 )。正椭圆算子的谱和周期双特征。发明。数学。29（1）：39 – 79。DOI：https://doi.org/10.1007/BF01405172。[Crossref]、[Web of Science®]、[  Google Scholar] ] 定理产生$○({\lambda }^{n-1})$在测地线流的一般条件下误差项的界限，我们也可以扩展 Bérard ( 1977) 贝拉尔，PH（1977 年）。关于无共轭点的紧黎曼流形上的波动方程。数学。Z. 155(3): 249 – 276。DOI：https://doi.org/10.1007/BF02028444。[Crossref]、[Web of Science®]、[  Google Scholar] ) 定理产生$哦({\lambda }^{n-1}/\hspace{0.17em}\text{日志}\hspace{0.17em}\lambda )$在假设主曲率处处非正的情况下的误差界限。如果我们加强对潜力的假设，我们可以获得对 tori 的进一步改进，这基本上是最优的$伏\in {升}^{磷}(米)$ 和 ${伏}^{-}\in \mathcal{钾}(米)$对于适当的指数p = p _n。