LAUSR

Let $$\Omega ={\widetilde{\Omega }}{\setminus } \overline{D}$$Ω=Ω~\D¯ where $${\widetilde{\Omega }}$$Ω~ is a bounded domain with connected complement in $${\mathbb {C}}^n$$Cn (or more generally in a Stein manifold) and D is relatively compact open subset of $${\widetilde{\Omega }}$$Ω~ with connected complement in $$\widetilde{\Omega }$$Ω~. We obtain characterizations of pseudoconvexity of $${\widetilde{\Omega }}$$Ω~ and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups of $$\Omega $$Ω on various function spaces. In particular, we show that if the boundaries of $${\widetilde{\Omega }}$$Ω~ and D are Lipschitz and $$C^2$$C2-smooth respectively, then both $${\widetilde{\Omega }}$$Ω~ and D are pseudoconvex if and only if 0 is not in the spectrum of the $$\overline{\partial }$$∂¯-Neumann Laplacian of $$\Omega $$Ω on (0, q)-forms for $$1\le q\le n-2$$1≤q≤n-2 when $$n\ge 3$$n≥3; or 0 is not a limit point of the spectrum of the $$\overline{\partial }$$∂¯-Neumann Laplacian on (0, 1)-forms when $$n=2$$n=2.