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Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients
Authors:Marco Cappiello
Institution:a Dipartimento di Matematica, Università di Ferrara, Via Machiavelli, 35, 44100 Ferrara, Italy
b Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
c Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Abstract:We show that all eigenfunctions of linear partial differential operators in Rn with polynomial coefficients of Shubin type are extended to entire functions in Cn of finite exponential type 2 and decay like exp(−2|z|) for |z|→∞ in conic neighbourhoods of the form |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 {zCn||Imz|?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
(∗)
Keywords:Harmonic oscillator  Shubin pseudodifferential operators  Gelfand-Shilov spaces
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