We study the asymptotic behavior of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott–Ruelle spectral data of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten–Helffer–Sjöstrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.

中文翻译：

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Pollicott–Ruelle谱和维滕拉普拉斯算子

我们研究了光滑无紧黎曼流形上Witten Laplacian特征值和特征模的渐近行为。我们证明它们收敛于作用在适当的各向异性Sobolev空间上的相应梯度流的Pollicott-Ruelle光谱数据。作为我们方法的应用，我们还构造了满足Witten-Helffer-Sjöstrand隧道公式和楔形产品Witten变形的Fukaya猜想的准准自然族。