I found this text in an article studying the Cauchy problem related to some PDE. where $\Omega \subset \mathbb{R}^3$ is bounded with smooth $\partial \Omega$ and $\alpha >0.$ I have to questions: 1.I wonder why the operator $(\alpha^2 \Delta^2 + \Delta)^{\frac 12}$ is bounded. The other eigenvalues $\lambda_j$ for which $(\alpha^2 \lambda_j^2 – \lambda_j)^{\frac 12}$ […]