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<p>In this article we prove that, over complete manifolds of dimension <i>n</i> with vanishing curvature at infinity, the essential spectrum of the Hodge Laplacian on differential <i>k</i>-forms is a connected interval for 0 ≤ <i>k</i> ≤ <i>n</i>. The main idea is to show that large balls of these manifolds, which capture their spectrum, are close in the Gromov–Hausdorff sense to product manifolds. We achieve this by carefully describing the collapsed limits of these balls. Then, via a new generalized version of the classical Weyl criterion, we demonstrate that very rough test forms that we get from the <i>ε</i>-approximation maps can be used to show that the essential spectrum is a connected interval. We also prove that, under a weaker condition where the Ricci curvature is asymptotically nonnegative, the essential spectrum on <i>k</i>-forms is [0, ∞), but only for 0 ≤ <i>k</i> ≤ <i>q</i> and <i>n</i> − <i>q</i> ≤ <i>k</i> ≤ <i>n</i> for some integer <i>q</i> ≥ 1 which depends on the structure of the manifolds at infinity. Our results can also be generalized to Schrödinger operators with double-well potential.</p>

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Connected essential spectrum: the case of differential forms

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  • 发表日期：

    2026年2月19日

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  • Nelia Charalambous,
  • Zhiqin Lu

摘要

In this article we prove that, over complete manifolds of dimension n with vanishing curvature at infinity, the essential spectrum of the Hodge Laplacian on differential k-forms is a connected interval for 0 ≤ k ≤ n. The main idea is to show that large balls of these manifolds, which capture their spectrum, are close in the Gromov–Hausdorff sense to product manifolds. We achieve this by carefully describing the collapsed limits of these balls. Then, via a new generalized version of the classical Weyl criterion, we demonstrate that very rough test forms that we get from the ε-approximation maps can be used to show that the essential spectrum is a connected interval. We also prove that, under a weaker condition where the Ricci curvature is asymptotically nonnegative, the essential spectrum on k-forms is [0, ∞), but only for 0 ≤ k ≤ q and n − q ≤ k ≤ n for some integer q ≥ 1 which depends on the structure of the manifolds at infinity. Our results can also be generalized to Schrödinger operators with double-well potential.