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| \rightarrow \infty $ in conic neighbourhoods of the form $|\Im z| \leq \gamma |\Re z |$. We also show that under semilinear polynomial perturbations all nonzero homoclinics %(if exist) keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip $\{z\in \C^n: \, |\Im z| \leq T\}$ for some $T>0$. The proofs are based on geometrical and perturbative methods in Gelfand--Shilov spaces. The results apply in particular to semilinear Schr\"{o}dinger equations of the form \begin{equation} \label{schro} -\Delta u + |x|^2u -\lambda u = F(x,u, \nabla u). \end{equation} Our estimates on homoclinics are sharp. In fact, we exhibit examples of solutions of \eqref{schro} with super-exponential decay, which are meromorphic functions, the key point of our argument being the celebrated great Picard theorem in complex analysis.
Super-exponential decay and holomorphic extensions for semilinar equations with polynomial coefficients
CAPPIELLO, Marco;
2006
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| \rightarrow \infty $ in conic neighbourhoods of the form $|\Im z| \leq \gamma |\Re z |$. We also show that under semilinear polynomial perturbations all nonzero homoclinics %(if exist) keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip $\{z\in \C^n: \, |\Im z| \leq T\}$ for some $T>0$. The proofs are based on geometrical and perturbative methods in Gelfand--Shilov spaces. The results apply in particular to semilinear Schr\"{o}dinger equations of the form \begin{equation} \label{schro} -\Delta u + |x|^2u -\lambda u = F(x,u, \nabla u). \end{equation} Our estimates on homoclinics are sharp. In fact, we exhibit examples of solutions of \eqref{schro} with super-exponential decay, which are meromorphic functions, the key point of our argument being the celebrated great Picard theorem in complex analysis.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
