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

We show that all eigenfunctions of linear partial differential operators in RnRn with polynomial coefficients of Shubin type are extended to entire functions in CnCn of finite exponential type 2 and decay like exp(−|z|2)exp(−|z|2) for |z|→∞|z|→∞ in conic neighbourhoods of the form |Imz|⩽γ|Rez||Imz|⩽γ|Rez|. 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∈Cn||Imz|⩽T}{z∈Cn||Imz|⩽T} for some T>0T>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 formequation(∗)−Δu+|x|2u−λu=F(x,u,∇u).−Δu+|x|2u−λu=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.