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

We show that all eigenfunctions of linear partial differential operators in $R^n$ with polynomial coefficients. 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. Our estimates on homoclinics are sharp. of Shubin type are extended to entire functions in $C^n$ of finite exponential type 2 and decay like $exp(−|z|2)$ for $|z|\to \infty$ in conic neighbourhoods of the form $|Im z| \leq |Re z|$.