Schrödinger-type equations in Gelfand-Shilov spaces

Abstract

We study the initial value problem for Schrödinger-type equations with initial data presenting a certain Gevrey regularity and an exponential behavior at infinity. We assume the lower order terms of the Schrödinger operator depending on (t, x) ∈ [0, T ] × R n and complex valued. Under a suitable decay condition as |x| → ∞ on the imaginary part of the first order term and an algebraic growth assumption on the real part, we derive global energy estimates in suitable Sobolev spaces of infinite order and prove a well posedness result in Gelfand-Shilov type spaces. We also discuss by examples the sharpness of the result.RÉSUMÉ. Nous étudions le problème de Cauchy pour des équations de Schrödinger avec des données initiales présentent une régularité de type Gevrey et un comportement exponentiel a l'infini. Nous supposons les coefficients des terms d'ordre inferieur dépendant de (t, x) ∈ [0, T ]×R n et à valeurs complexes. Sous une condition de décroissance convenable pour |x| → ∞ pour la partie imaginaire du terme du premier ordre et une hypotèse de croissance algébrique sur la partie réelle, nous obtenons des estimations de l'énergie globales dans des espaces de Sobolev d'ordre infini et prouvons que le problème de Cauchy est bien posé dans des espaces de type Gelfand-Shilov. Nous discutons également par des exemples de l'optimalité du résultat.

The interplay between decay of the data and regularity of the solution in Schrödinger equations

“…This result is consistent with the one obtained in [2] for the critical case s = 1 1−σ . We can so overcome the critical index 1/(1 − σ ) for G s well-posedness by allowing a suitable loss of asymptotic behavior as |x| → ∞ in the used weights.…”

“…We remark that, in comparison with [2] in the case of uniformly bounded in x coefficients and in comparison with [6], we obtain by Corollary 1 a Sobolev solution valued in H m loc without any assumption on the spatial derivatives of a j . Furthermore, in comparison with [6], where a pointwise estimate for u is given with a time-dependent constant tending to infinity for t → +0, we have to mention that here, since we do not look for smoothing, we obtain for the solution u an energy estimate on the whole interval [0, T * ].…”

“…So, if u ∈ A s 1 ,s 2 A,ρ (H m ) with A and ρ large enough, then e Λ u ∈ H m . Similarly, one obtains that for every u ∈ H m the function R e −Λ u ∈ B s 1 ,s 2 A,0 (H m ) has the property…”

“…We recall that the SG (Symbol Global) calculus requires symbol like behavior of the coefficients also with respect to the spatial variables. -A partial answer to question Q1 has been given, again in the framework of the SG calculus, as a by-product of [2]. Under the assumption…”

“…Finally, the unique solution u = (e Λ ) −1 v of the original Cauchy problem satisfies e (A−4M) x 1−σ u(t, •) ∈ H m for every t ∈ [0, T ]. For this reason, u(t, •) may belong to a weighted Sobolev space with exponentially decreasing weight, compare with [2,6]. Finally, the solution u satisfies the following energy estimate:…”