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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[16] Sogge, Christopher D. Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 105, Cambridge University Press, Cambridge, 1993 | DOI | MR | Zbl

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

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