Asymptotic behaviour of the fundamental solution of a second-order parabolic equation

We study the asymptotic behaviour as $t\to\infty$ of the fundamental solution (FS) $G(x,s,t)$ of the Cauchy problem for the parabolic equation $G_t-G_{xx}+a(x)G=0$, $x\in{\mathbb R}^1$, $t>0$. We suppose that the coefficient $a(x)$ can be written as $x\to\pm\infty$ in the form $a(x)=a_2^\pm x^{-2}+\varphi (x)$, where the function $\phi(x)$ has an asymptotic expansion as $x\to\pm\infty$ in positive powers of $x^{-1}$ and $|\varphi (x)|=o(|x|^{-2})$. We construct and justify the asymptotic expansion of the FS $G(z,s,t)$ as $t\to\infty$ up to any power of $t^{-1}$ for the whole plane $x,s\in{\mathbb R}^1$.