论文标题
具有奇异潜力的尖锐的Weyl定律
Sharp Weyl laws with singular potentials
论文作者
论文摘要
我们认为在三维的riemannian流形中，拉普拉斯 - 贝特拉米操作员受到加藤类潜力的扰动，并研究了在这种扰动下的各种形式的Weyl定律是否有效。我们表明，对于具有标准尖锐剩余期限的任何Kato类潜力，Weyl Law的定义是通过附加术语修改的。额外的任期始终比领先期限较低，但是它可能比较低的剩余期限较低。特别是，我们提供了奇异潜力的例子，该术语违反了标准拉普拉斯 - 贝特拉米操作员的尖锐的Weyl定律。为了证明，我们将Avakumović的方法扩展到具有奇异潜力的Schrödinger经营者的情况。
We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumović to the case of Schrödinger operators with singular potentials.
