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)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 3, page 1091-1121
  • ISSN: 0373-0956

Abstract

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We consider Schrödinger operators H on ℝ n with variable coefficients. Let H 0 = - 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 - i t H can be written as a product of the free evolution operator e - i t H 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.

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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  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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