According Dehman, Gérard and Lebeau in "Stabilization and control for the nonlinear Schrödinger equation on a compact surface" (Math. Z. (2006) 254:729–749, DOI 10.1007/s00209-006-0005-3) is claimed that the operator $$Jv=v+i\,a(x)(I-\Delta)^{-1}\,v$$ is an isomorphism on $H^s$ and $L^p$, $s\in\,\mathbb{R}$ and $1\leq p\leq +\infty$ where $$a\in\,C^\infty(\Omega)\,\cap\,W^{1,\infty}(\Omega).$$
My question: the operator $J^{-1}$ is continuous of $L^2(\Omega)$ in $L^2(\Omega)?$ In affirmative case, how am I prove this?