具有奇异势的尖锐 Weyl 定律
Sharp Weyl laws with singular potentials
AI总结：
本文研究三维 Riemann 流形上受 Kato 类奇异势扰动的 Laplace–Beltrami 算子，推广 Avakumović 方法，证明带附加项和尖锐余项的逐点 Weyl 定律，并构造反例说明附加项可破坏标准尖锐定律。
摘要：
我们考虑三维 Riemann 流形上的 Laplace–Beltrami 算子受到一个来自 Kato 类的势扰动，并研究在这种扰动下各种形式的 Weyl 定律是否仍然成立。我们证明，对于任何 Kato 类势，逐点 Weyl 定律成立，但会增加一个附加项，并带有标准的尖锐余项。该附加项相对于主项始终是低阶的，但相对于尖锐余项则可能是低阶的，也可能不是。特别地，我们给出了若干奇异势的例子，在这些例子中，该附加项破坏了标准 Laplace–Beltrami 算子的尖锐逐点 Weyl 定律。在证明中，我们将 Avakumović 方法推广到具有奇异势的 Schrödinger 算子情形。
英文摘要：
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.
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