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# Asymptotic properties of a functional defined on Sobolev space

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 !

1. 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