On eigenfunctions and nodal sets of the Witten-Laplacian

TL;DR

This paper extends classical eigenfunction theory to the Witten-Laplacian, establishing a Courant-type theorem, analyzing nodal lines on Riemannian surfaces, and providing bounds on eigenvalue multiplicities.

Contribution

It introduces a Courant-type nodal domain theorem for the Witten-Laplacian and characterizes nodal lines on Riemannian surfaces, with bounds on eigenvalue multiplicities.

Findings

Established a Courant-type nodal domain theorem for the Witten-Laplacian.

Characterized properties of nodal lines on smooth Riemannian 2-manifolds.

Provided an upper bound for the multiplicity of closed eigenvalues 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.