Let $(\lambda_k,\phi_k)$ be the eigenvalues/eigenvectors of the Laplacian operator ($-\Delta$) on a smooth and bounded domain $\Omega \subset \mathbb{R^n}$. From the spectral theorem in Hilbert spaces we know that the eigenvectors form a complete orthonormal basis for $L^2(\Omega)$. Thus for $u \in L^2(\Omega)$ we may write $$ u = \sum_k \langle u,\phi_k\rangle\ \phi_k \quad \text{and} \quad -\Delta u = \sum_k \lambda_k\langle u,\phi_k\rangle\ \phi_k $$ In this setting, I have read that we can define fractional powers of the Laplacian as $$ -\Delta^s u = \sum_k \lambda_k^s \langle u,\phi_k\rangle\ \phi_k \quad,\quad s \in \mathbb{R} \tag{$\star$} $$ Questions
- But, is ($\star$) valid for every $s \in \mathbb{R}$? Isn't the above sum diverging for certain $s \in \mathbb{R}$? Under what conditions?
- Is it true that we can find any pair $(\lambda_k,\phi_k)$ corresponding to a fractional Laplacian operator in $\Omega$ using only ($\star$) and the eigenvectors/eigenvalues of the Laplacian ($s=1$)?
- Where should I look for the regularity theory surrounding fractional Laplacians?
Pardon me for the multiple questions, but I am also asking for a reference for further study.
Thanks in advance!