On eigenfunctions and nodal sets of the Witten-Laplacian

In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal lines of the eigenfunctions of the Witten-Laplacian on smooth Riemannian $2$-manifolds. Besides, for a Riemann surface with genus $g$, an upper bound for the multiplicity of closed eigenvalues of the Witten-Laplacian can be provided.

💡 Research Summary

The paper studies spectral properties of the Witten‑Laplacian (\Delta_{\phi}=\Delta-\nabla\phi\cdot\nabla) on smooth Riemannian manifolds, focusing on two eigenvalue problems: the Dirichlet problem on a bounded domain (\Omega) and the closed (global) problem on a complete manifold (M). Because (\Delta_{\phi}) is self‑adjoint with respect to the weighted volume form (d\eta=e^{-\phi}dv), its spectrum is discrete and consists of a non‑decreasing sequence of eigenvalues. The authors first recall the variational characterizations of these eigenvalues (Rayleigh quotients) and note that the associated eigenfunctions form a complete orthogonal basis in (L^{2}(\Omega,d\eta)) (or (L^{2}(M,d\eta)) for the closed case).

The main contribution is a Courant‑type nodal domain theorem for the Witten‑Laplacian. For the Dirichlet problem, they prove that the (k)-th eigenfunction has at most (k) nodal domains. The proof follows the classical Courant argument: assuming the nodal domains are normal (i.e., open, connected components of the complement of the zero set), they restrict the eigenfunction to each domain, construct a linear combination that is orthogonal to the first (k-1) eigenfunctions, and compare its Rayleigh quotient with (\lambda_{k,\phi}). Equality forces the combination to be an eigenfunction supported on exactly (k) domains; any extra domain would lead to a contradiction via the maximum principle.

For the closed eigenvalue problem, the lack of a boundary prevents a direct use of the divergence theorem. The authors therefore establish an integral identity (Lemma 2.1) for any nodal domain (D): \