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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

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