In this paper, 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 $\mathbb R^d$, $(-\Delta)$ is the Dirichlet Laplacian in $\Omega$, 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 $\theta=1-\frac s m$?

More precisely, my question is: How can I characterize $\text{dom}(-\Delta)^{\frac s 2}$ in terms of Sobolev spaces?