Abstract. Let $\Omega$ be a domain in $\mathbb C^n$ of the form $\Omega = \tilde \Omega \setminus \bar D$ where $\tilde \Omega$ is a bounded domain with connected complement and $D$ is a relatively compact open subset of $\tilde \Omega$ with connected complement in $\tilde \Omega$ . In this talk, we will explain how one can detect pseudoconvexity of $\tilde \Omega$ and $D$ through spectral property of the $\bar \partial$-Neumann Laplacian. In particular, we show that if the boundary $b\tilde \Omega$ is Lipschitz and $bD$ is $\mathcal C^2$-smooth, then both $\tilde \Omega$ and $D$ are pseudoconvex if and only if $0$ is not in the spectrum of the $\bar \partial$-Neumann Laplacian on $(0, q)$-forms for $1 \leq q \leq n − 2$ when $n \geq 3$; or $0$ is not a limit point for the spectrum of the $\bar \partial$-Neumannn Laplacian on $(0, 1)$-forms when $n = 2$. This is a joint work with Christine Laurent-Thiébaut and Mei-Chi Shaw.