In [this paper][1], the authors claim that for $s\in [0,1]$, $\left [ H_0^1(\Omega), L^2(\Omega)\right ]_\theta=\text{dom}(-\Delta)^{\frac s 2}$, where $\Omega$ is a smooth and bounded domain in $\mathbb R^d$, and $\theta =1-s$. Here, the interpolation theory is taken from the book of Lions and Magenes "Problèmes aux limites non homogènes et applications", Vol 1. Is the following more general statement true:
$$\left [ H_0^m(\Omega), L^2(\Omega)\right ]_\theta=\text{dom}(-\Delta)^{\frac s 2}$$
for $m=\lfloor s \rfloor +1$, and $\theta=1-\frac s m$?


  [1]: https://arxiv.org/pdf/1302.0698.pdf