Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

Abstract

We show that all eigenfunctions of linear partial differential operators in R n with polynomial coefficients of Shubin type are extended to entire functions in C n of finite exponential type 2 and decay like exp ( − | z | 2 ) for | z | → ∞ in conic neighbourhoods of the form | Im z | ⩽ γ | Re z | . We also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip { z ∈ C n | | Im z | ⩽ T } for some T > 0 . The proofs are based on geometrical and perturbative methods in Gelfand–Shilov spaces. The results apply in particular to semilinear Schrödinger equations of the form (∗) − Δ u + | x | 2 u − λ u = F ( x , u , ∇ u ) . Our estimates on homoclinics are sharp. In fact, we exhibit examples of solutions of (∗) with super-exponential decay, which are meromorphic functions, the key point of our argument being the celebrated great Picard theorem in complex analysis.