Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients
By Marco Cappiello, Todor Gramchev, L. Rodino. 2006
48 citations recorded by the source corpus.
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 | → ∞ in conic neighbourhoods of the form | Im z | ⩽ γ | Re z | . 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 ∈ C n | | Im z | ⩽ 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 Schrodinger equations of the form (∗) − Δ u + | x | 2 u − λ u = 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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2f8a7cc3b499610033e894ef07b87dc5979c4e40
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In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Entire extensions and exponential decay for semilinear elliptic equations (2010, methodology): “Generalizing the arguments in [7] we consider the equation where h, (cid:96) ∈ Z + , (cid:96) > 1 , h > 1 , h odd.”