Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Ito, Kenichi; Nakamura, Shu

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1091-1121.

Nous considérons des opérateurs de Schrödinger $H$ à coefficients variables sur ${ℝ}^n$, qui sont des perturbations “à courte portée” de l’opérateur de Schrödinger libre $H_0=-\frac{1}{2}▵$. Dans le cas non captant, nous montrons que l’opérateur d’évolution temporelle $e^{-itH}$ s’écrit comme le produit de l’opérateur d’évolution libre $e^{-it{H}_0}$ et d’un opérateur intégral de Fourier $W\left(t\right)$, qui est associé à la relation canonique donnée par la diffusion classique. Nous établissons aussi un résultat similaire pour les opérateurs d’onde. Ces résultats sont analogues à ceux obtenus par Hassell et Wunsch, mais leurs hypothèses, leur preuve et leur formulation sont nettement différents. La preuve repose sur un théorème de type Egorov semblable à ceux utilisés dans les travaux précédents des auteurs, et qui est combiné ici à une caractérisation de type Beals des opérateurs intégraux de Fourier.

We consider Schrödinger operators $H$ on ${ℝ}^n$ with variable coefficients. Let $H_0=-\frac{1}{2}▵$ be the free Schrödinger operator and we suppose $H$ is a “short-range” perturbation of $H_0$. Then, under the nontrapping condition, we show that the time evolution operator: $e^{-itH}$ can be written as a product of the free evolution operator $e^{-it{H}_0}$ and a Fourier integral operator $W\left(t\right)$ which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results are analogous to results by Hassell and Wunsch, but the assumptions, the proof and the formulation of results are considerably different. The proof employs an Egorov-type theorem similar to those used in previous works by the authors combined with a Beals-type characterization of Fourier integral operators.

Classification : 35Q40, 35A17, 35A21
Mots clés : Schrödinger equation, fundamental solutions, scattering theory

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