On the asymptotics of the fundamental solution of a parabolic equation in the critical case

The behavior as $t\to\infty$ of the fundamental solution $G(x,s,t)$ of the Cauchy problem for the equation $u_t=u_{xx}-a(x)u$ $(x\in\mathbf R^1$, $t>0)$ is studied in the case when the decay rate of the coefficient $a(x)$ as $x\to\pm\infty$ is critical:
$$ a(x)=a_2^\pm x^{-2}+\sum_{i=3}^\infty a_i^\pm x^{-i}\qquad(x\to\pm\infty). $$
The asymptotic expansion of $G(x,s,t)$ as $t\to\infty$ is constructed and established for all $x,s\in\mathbf R^1$. The fundamental solution decays like a power, and the decay rate is determined by the quantities $a_2^\pm$.
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