Open Access
Translator Disclaimer
April, 2013 Resolvent estimates in amalgam spaces and asymptotic expansions for Schrödinger equations
Artbazar GALTBAYAR, Kenji YAJIMA
J. Math. Soc. Japan 65(2): 563-605 (April, 2013). DOI: 10.2969/jmsj/06520563

Abstract

We consider Schrödinger equations $i\partial_t u = (-\Delta + V)u$ in ${\mathbb R}^3$ with a real potential $V$ such that, for an integer $k\geq 0$, $\langle x \rangle^k V(x)$ belongs to an amalgam space $\ell^p(L^q)$ for some $1\leq p$ < 3/2 < $q \leq \infty$, where $\langle x \rangle=(1+|x|^2)^{1/2}$. Let $H = -\Delta + V$ and let $P_{ac}$ be the projector onto the absolutely continuous subspace of $L^2({\mathbb R}^3)$ for $H$. Assuming that zero is not an eigenvalue nor a resonance of $H$, we show that solutions $u(t) = \exp(-itH)P_{ac}\varphi$ admit asymptotic expansions as $t \to \infty$ of the form

$$\bigg\| \langle x \rangle^{-k-\varepsilon} \bigg( u(t)- \sum_{j=0}^{[k/2]}t^{-\frac32-j}P_j \varphi \bigg) \bigg\|_{\infty} \leq C |t|^{-\frac{k+3+\varepsilon}2} \big\| \langle x \rangle^{k+\varepsilon}\varphi \big\|_1$$

for 0 < $\varepsilon$ < $3(1/p-2/3)$, where $P_0, \dots, P_{[k/2]}$ are operators of finite rank and $[k/2]$ is the integral part of $k/2$. The proof is based upon estimates of boundary values on the reals of the resolvent $(-\Delta -\lambda^2)^{-1}$ as an operator-valued function between certain weighted amalgam spaces.

Citation

Download Citation

Artbazar GALTBAYAR. Kenji YAJIMA. "Resolvent estimates in amalgam spaces and asymptotic expansions for Schrödinger equations." J. Math. Soc. Japan 65 (2) 563 - 605, April, 2013. https://doi.org/10.2969/jmsj/06520563

Information

Published: April, 2013
First available in Project Euclid: 25 April 2013

zbMATH: 1284.35361
MathSciNet: MR3055596
Digital Object Identifier: 10.2969/jmsj/06520563

Subjects:
Primary: 35Q41

Keywords: asymptotic expansions , resolvent estimates , Schrödinger equation

Rights: Copyright © 2013 Mathematical Society of Japan

JOURNAL ARTICLE
43 PAGES


SHARE
Vol.65 • No. 2 • April, 2013
Back to Top