In this talk I consider generalized eigenvalue problems for a family of operators with a polynomial dependence on a complex parameter. The basic example is the following equation −∆u+(P(x)−λ)2u=0, where P is a polynomial of degree m≥2 such that the homogeneous part Pm​ of P satisfies Pm​(x)>0 for every $x ∈ R^n\\{0\}$, λ∈C and u is in the space L2(Rn). The problem is to find solutions λ and u for this equation. It is equivalent to a genuine non self-adjoint operator for a 2×2 system. We discuss here existence of non trivial eigenstates (λ,u) for models coming from analytic theory of smoothness for PDE. I shall review some old results and presents recente improvements on this subject.