Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Ito, Kenichi ; Nakamura, Shu

Annales de l'Institut Fourier, Tome 62 (2012), p. 1091-1121 / Harvested from Numdam

Résumé

Nous considérons des opérateurs de Schrödinger $H$ à coefficients variables sur $\mathbb{R}^n$, qui sont des perturbations “à courte portée” de l’opérateur de Schrödinger libre $H_0=-\frac{1}{2} \triangle $. 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 $e^{-itH_0}$ et d’un opérateur intégral de Fourier $W(t)$, 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.

Abstract

We consider Schrödinger operators $H$ on $\mathbb{R}^n$ with variable coefficients. Let $H_0=-\frac{1}{2}\triangle $ 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 $e^{-itH_0}$ and a Fourier integral operator $W(t)$ 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.

Détail

Publié le : 2012-01-01

MR 3013818 | Zbl 1251.35102

DOI : https://doi.org/10.5802/aif.2718
Classification:  35Q40,  35A17,  35A21

Bibliographie

[1] Craig, Walter; Kappeler, Thomas; Strauss, Walter Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math., Tome 48 (1995) no. 8, pp. 769-860 | Article | MR 1361016 | Zbl 0856.35106

[2] Fujiwara, Daisuke Remarks on convergence of the Feynman path integrals, Duke Math. J., Tome 47 (1980) no. 3, pp. 559-600 http://projecteuclid.org/getRecord?id=euclid.dmj/1077314181 | Article | MR 587166 | Zbl 0457.35026

[3] Hassell, Andrew; Wunsch, Jared On the structure of the Schrödinger propagator, Partial differential equations and inverse problems, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 362 (2004), pp. 199-209 | MR 2091660

[4] Hassell, Andrew; Wunsch, Jared The Schrödinger propagator for scattering metrics, Ann. of Math. (2), Tome 162 (2005) no. 1, pp. 487-523 | Article | MR 2178967

[5] Hörmander, Lars Fourier integral operators. I, Acta Math., Tome 127 (1971) no. 1-2, pp. 79-183 | Article | MR 388463 | Zbl 0212.46601

[6] Hörmander, Lars The analysis of linear partial differential operators. I–IV, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (1983–1985) (Fourier integral operators) | MR 717035 | Zbl 0521.35001

[7] Ito, Kenichi Propagation of singularities for Schrödinger equations on the Euclidean space with a scattering metric, Comm. Partial Differential Equations, Tome 31 (2006) no. 10-12, pp. 1735-1777 | Article | MR 2273972

[8] Ito, Kenichi; Nakamura, Shu Singularities of solutions to the Schrödinger equation on scattering manifold, Amer. J. Math., Tome 131 (2009) no. 6, pp. 1835-1865 | Article | MR 2567509

[9] Kapitanski, L.; Safarov, Yu. A parametrix for the nonstationary Schrödinger equation, Differential operators and spectral theory, Amer. Math. Soc., Providence, RI (Amer. Math. Soc. Transl. Ser. 2) Tome 189 (1999), pp. 139-148 | MR 1730509 | Zbl 0922.35144

[10] Martinez, André; Nakamura, Shu; Sordoni, Vania Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math., Tome 59 (2006) no. 9, pp. 1330-1351 | Article | MR 2237289

[11] Martinez, André; Nakamura, Shu; Sordoni, Vania Analytic wave front set for solutions to Schrödinger equations, Adv. Math., Tome 222 (2009) no. 4, pp. 1277-1307 | Article | MR 2554936

[12] Nakamura, Shu Propagation of the homogeneous wave front set for Schrödinger equations, Duke Math. J., Tome 126 (2005) no. 2, pp. 349-367 | Article | MR 2115261

[13] Nakamura, Shu Semiclassical singularities propagation property for Schrödinger equations, J. Math. Soc. Japan, Tome 61 (2009) no. 1, pp. 177-211 http://projecteuclid.org/getRecord?id=euclid.jmsj/1234189032 | Article | MR 2272875

[14] Nakamura, Shu Wave front set for solutions to Schrödinger equations, J. Funct. Anal., Tome 256 (2009) no. 4, pp. 1299-1309 | Article | MR 2488342

[15] Robbiano, Luc; Zuily, Claude Microlocal analytic smoothing effect for the Schrödinger equation, Duke Math. J., Tome 100 (1999) no. 1, pp. 93-129 | Article | MR 1714756 | Zbl 0941.35014

[16] Sogge, Christopher D. Fourier integrals in classical analysis, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 105 (1993) | Article | MR 1205579 | Zbl 0783.35001

[17] Wunsch, Jared Propagation of singularities and growth for Schrödinger operators, Duke Math. J., Tome 98 (1999) no. 1, pp. 137-186 | Article | MR 1687567 | Zbl 0953.35121

[18] Yajima, Kenji Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations, Comm. Math. Phys., Tome 181 (1996) no. 3, pp. 605-629 http://projecteuclid.org/getRecord?id=euclid.cmp/1104287904 | Article | MR 1414302 | Zbl 0883.35022