Singular perturbations of powers of the Laplacian on the torus (Q2436394)

In this short note, the author deals with some perturbation \(S\) of the fractional Laplacian on the torus \(\mathbb{T}^n\), given by \[ S=(-\Delta)^\alpha + Q, \] where \(Q\) is a linear operator, by reviewing some results related to the embedding of Bessel potential spaces in multiplier spaces. The author states several results: such singularly perturbed operator \(S\) is sectorial (and, in some cases, self-adjoint and bounded) in \(L^2(\mathbb{T}^n)\); sequences of operators \(S_k\) converge uniformly in some multiplier space; the corresponding eigenfunctions system is complete in \(L^2(\mathbb{T}^n)\); and so on.