# 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

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topIto, 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 -

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