The study of a new family of highly non-self-adjoint operators which is reminiscent of the classical Toeplitz operators arises naturally in the context of evolution problems connected to sandpiles/slow-fast diffusion. The aim of this talk is to describe the difficulties involved in determining general spectral properties of operators in this family and show how to overcome these difficulties. In turns we will establish basisness/non-basisness theorems for general families of 1D periodic functions. We will then apply the latter, in order to examine non-orthogonal projection methods for the $p$-poisson parabolic time-evolution initial value problem with stochastic forcing.
