I am currently reading the paper "Variational Autoencoding Neural Operators", which performs experiments on 1D gaussian random fields (i.e., gaussian processes).
They generate functions according to the following process.
"A zero-mean Gaussian random field on $X = [0,1]$, with zero boundary conditions and covariance operator $\Gamma = (I - \Delta)^{-\alpha}$. This operator admits the orthonormal eigendecomposition $\Gamma = \sum_{I=1}^\infty \lambda_i \phi_i \otimes \phi_i,$ where $\lambda_i = ((2 \pi i)^2 + \tau^2)^{-\alpha}$ and $\phi_i(x) = \sqrt{2} \sin\left(2 \pi i x\right)$".
They use the Karhunen-Loéve theorem to construct the random functions from this process, which I understand fine, but my question is concerned with this first part.
How can I obtain the eigendecomposition of this process defined by such covariance operator? Are there some resources to learn about processes of such kind ($(I-\Delta)^{-\alpha}$ covariance)? I struggle to find relevant posts or articles precisely about that.