On eigenfunctions and nodal sets of the Witten-Laplacian

TL;DR

This work extends Courant’s nodal domain theorem to the Witten-Laplacian $\Delta_{\phi}$ on bounded domains and in the closed setting, using a weighted measure $d\eta=e^{-\phi}dv$ and a self-adjoint variational framework. It develops a two-case strategy for proving nodal-domain bounds, combining Rayleigh quotient arguments with integral Green’s formulas and limiting processes to link eigenvalues to first Dirichlet eigenvalues on nodal domains. The paper also characterizes the nodal lines on smooth $2$-manifolds via local regularity results (Bers) and Aronszajn-type arguments, obtaining a genus-dependent bound on closed-eigenvalue multiplicities $\mathrm{mult}(\lambda^{c}_{i,\phi}(M^2))\le (2g+i+1)(2g+i+2)/2$. These results advance spectral geometry for weighted Laplacians and have implications for the structure of nodal sets and eigenvalue multiplicities on Riemann surfaces.

Abstract

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.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof