Asymptotic Expansions of Solutions to the Heat Equations with Initial Value in the Dual of Gel'fand-Shilov Spaces

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.