# Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Kenichi Ito[1]; Shu Nakamura[2]

• [1] University of Tsukuba Graduate School of Pure and Applied Sciences 1-1-1 Tennodai, Tsukuba Ibaraki, 305-8571 (Japan)
• [2] University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba, Meguro Tokyo, 153-8914 (Japan)
• Volume: 62, Issue: 3, page 1091-1121
• ISSN: 0373-0956

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## Abstract

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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.

## How to cite

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Ito, Kenichi, and Nakamura, Shu. "Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients." Annales de l’institut Fourier 62.3 (2012): 1091-1121. <http://eudml.org/doc/251085>.

@article{Ito2012,
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.},
affiliation = {University of Tsukuba Graduate School of Pure and Applied Sciences 1-1-1 Tennodai, Tsukuba Ibaraki, 305-8571 (Japan); University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba, Meguro Tokyo, 153-8914 (Japan)},
author = {Ito, Kenichi, Nakamura, Shu},
journal = {Annales de l’institut Fourier},
keywords = {Schrödinger equation; fundamental solutions; scattering theory},
language = {eng},
number = {3},
pages = {1091-1121},
publisher = {Association des Annales de l’institut Fourier},
title = {Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients},
url = {http://eudml.org/doc/251085},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Ito, Kenichi
AU - Nakamura, Shu
TI - Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 3
SP - 1091
EP - 1121
AB - 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.
LA - eng
KW - Schrödinger equation; fundamental solutions; scattering theory
UR - http://eudml.org/doc/251085
ER -

## References

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1. Walter Craig, Thomas Kappeler, Walter Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1995), 769-860 Zbl0856.35106MR1361016
2. Daisuke Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J. 47 (1980), 559-600 Zbl0457.35026MR587166
3. Andrew Hassell, Jared Wunsch, On the structure of the Schrödinger propagator, Partial differential equations and inverse problems 362 (2004), 199-209, Amer. Math. Soc., Providence, RI Zbl1062.35098MR2091660
4. Andrew Hassell, Jared Wunsch, The Schrödinger propagator for scattering metrics, Ann. of Math. (2) 162 (2005), 487-523 Zbl1126.58016MR2178967
5. Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79-183 Zbl0212.46601MR388463
6. Lars Hörmander, The analysis of linear partial differential operators. I–IV, (1983–1985), Springer-Verlag, Berlin Zbl0521.35001MR717035
7. Kenichi Ito, Propagation of singularities for Schrödinger equations on the Euclidean space with a scattering metric, Comm. Partial Differential Equations 31 (2006), 1735-1777 Zbl1117.35005MR2273972
8. Kenichi Ito, Shu Nakamura, Singularities of solutions to the Schrödinger equation on scattering manifold, Amer. J. Math. 131 (2009), 1835-1865 Zbl1186.35004MR2567509
9. L. Kapitanski, Yu. Safarov, A parametrix for the nonstationary Schrödinger equation, Differential operators and spectral theory 189 (1999), 139-148, Amer. Math. Soc., Providence, RI Zbl0922.35144MR1730509
10. André Martinez, Shu Nakamura, Vania Sordoni, Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. 59 (2006), 1330-1351 Zbl1122.35027MR2237289
11. André Martinez, Shu Nakamura, Vania Sordoni, Analytic wave front set for solutions to Schrödinger equations, Adv. Math. 222 (2009), 1277-1307 Zbl1180.35016MR2554936
12. Shu Nakamura, Propagation of the homogeneous wave front set for Schrödinger equations, Duke Math. J. 126 (2005), 349-367 Zbl1130.35023MR2115261
13. Shu Nakamura, Semiclassical singularities propagation property for Schrödinger equations, J. Math. Soc. Japan 61 (2009), 177-211 Zbl1176.35045MR2272875
14. Shu Nakamura, Wave front set for solutions to Schrödinger equations, J. Funct. Anal. 256 (2009), 1299-1309 Zbl1155.35014MR2488342
15. Luc Robbiano, Claude Zuily, Microlocal analytic smoothing effect for the Schrödinger equation, Duke Math. J. 100 (1999), 93-129 Zbl0941.35014MR1714756
16. Christopher D. Sogge, Fourier integrals in classical analysis, 105 (1993), Cambridge University Press, Cambridge Zbl0783.35001MR1205579
17. Jared Wunsch, Propagation of singularities and growth for Schrödinger operators, Duke Math. J. 98 (1999), 137-186 Zbl0953.35121MR1687567
18. Kenji Yajima, Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations, Comm. Math. Phys. 181 (1996), 605-629 Zbl0883.35022MR1414302

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