We study the heat trace for both the drifting Laplacian as well as Schrodinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. We also demonstrate that the more precise asymptotic behavior of the remainder is determined by and conversely distinguishes higher (Sobolev) regularity on the potential or weight function. In the case of a smooth weight function, we determine the full asymptotic expansion of the heat trace for the drifting Laplacian for small times. We then use the heat trace to study the asymptotics of the eigenvalue counting function. In both cases the Weyl law coincides with the Weyl law for the Riemannian manifold with the standard Laplace-Beltrami operator. We conclude by demonstrating isospectrality results for the drifting Laplacian on compact manifolds.

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流形上漂移的拉普拉斯算子和薛定谔算子的热迹

我们研究了紧致黎曼流形上漂移的拉普拉斯算子和薛定谔算子的热迹。在有限正则势或权函数的情况下，我们证明了热迹的部分（六项）渐近扩展的存在以及合适的余数估计。我们还证明了余数的更精确的渐近行为由势函数或权重函数的更高（Sobolev）正则性决定并相反地区分。在平滑权重函数的情况下，我们确定了小时间漂移的拉普拉斯算子的热迹的完全渐近扩展。然后我们使用热迹来研究特征值计数函数的渐近性。在这两种情况下，外尔定律与使用标准拉普拉斯-贝尔特拉米算子的黎曼流形的外尔定律一致。我们通过证明紧凑流形上漂移的拉普拉斯算子的等谱结果得出结论。