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 $(\alpha^2 \lambda_j^2 – \lambda_j)^{\frac 12}$

is non-negative are infinite and unbounded !

- Consider the operators $$ (\alpha^2 \Delta^2 + \Delta)^{-\frac 12}\,{\rm e}^{\alpha \Delta + (-1)^k (\alpha^2 \Delta + \Delta)^{\frac 12}},$$

let $u\in H^2(\mathbb{R^3})$ be in the domain of the above operator and define the functions $f_k:]0, 1] \to H^2(\mathbb{R^3})$

$$f_k(\alpha) := (\alpha^2 \Delta^2 + \Delta)^{-\frac 12} {\rm e}^{\alpha \Delta + (-1)^k (\alpha^2 \Delta^2 + \Delta)^{\frac 12}} u.$$

I’d like to prove (if possible) the existence of a function $\epsilon$ satisfying

$$\lim_{x\to 0} \epsilon(x) = 0$$ such that

$$\|f_k(\alpha) – f_k(\beta)\|_{H^2} \leq \epsilon(|\alpha – \beta|).$$

I wonder if it is possible (in some sense) to use the mean value theorem to finde

a suitable estimate for $\|f_k(\alpha) – f_k(\beta)\|_{H^2}.$

Thank you for any hint