On necessary and sufficient conditions for the convergence of solutions of one-dimensional diffusion stochastic equations with a non-regular dependence of coefficients on a parameter

We consider an one-dimensional stochastic differential equation of diffusion type
$$ d\xi_\alpha(t)=a_\alpha(\xi_\alpha(t))\,dt+\sigma_\alpha(\xi_\alpha(t))\,dw_\alpha(t),\qquad t>0. $$
where $\alpha>0$ is a parameter, $a_\alpha(x)$, $\sigma_\alpha(x)>0$ are real functions which may degenerate at some points $x_k$ as $\alpha\to 0$ and $w_\alpha(t)$ is a family of Wiener processes. The necessary and sufficient conditions for the weak convergence of $\xi_\alpha(t)$ to the generalized diffusion process $\alpha\to 0$ are obtained.