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December 2011 Asymptotic Expansions of Solutions to the Heat Equations with Initial Value in the Dual of Gel'fand-Shilov Spaces
Yasuyuki OKA
Tokyo J. Math. 34(2): 547-567 (December 2011). DOI: 10.3836/tjm/1327931402

Abstract

We will derive the asymptotic expansions of the solutions $U(x,t)$ to the heat equation with $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$, $r\geq 1/2$, initial value, where $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ is the dual space of the Gel'fand-Shilov space $\mathcal{S}^r_r(\mathbf{R}^d$. Moreover, we show that, when $1/2\leq r\leq 1$, these asymptotic expansions satisfy the strong asymptotic condition on some circle $D_R=\{t\in\mathbf{C}\ |\ \mathrm{Re}\ t^{-1}>R^{-1}\}$. Therefore, we find that these asymptotic series for $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ initial value are Borel summable by means of A. D. Sokal's result on the Borel summability. As an application, we show the asymptotic expansions of the Weyl transform with Planck's constant $\hbar$ in some state, which are refinement of a classical limit of the quantum mechanical expectation values expressed by the Weyl transform.

Citation

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Yasuyuki OKA. "Asymptotic Expansions of Solutions to the Heat Equations with Initial Value in the Dual of Gel'fand-Shilov Spaces." Tokyo J. Math. 34 (2) 547 - 567, December 2011. https://doi.org/10.3836/tjm/1327931402

Information

Published: December 2011
First available in Project Euclid: 30 January 2012

zbMATH: 1242.46050
MathSciNet: MR2918922
Digital Object Identifier: 10.3836/tjm/1327931402

Subjects:
Primary: 46F05
Secondary: 46F15, 81S30

Rights: Copyright © 2011 Publication Committee for the Tokyo Journal of Mathematics

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Vol.34 • No. 2 • December 2011
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