Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593013 | Journal of Functional Analysis | 2006 | 21 Pages |
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.