  In this paper, we study the approximation problem for functions in the
Gaussian-weighted Sobolev space $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed
smoothness $\alpha \in \mathbb{N}$ with error measured in the Gaussian-weighted
space $L_q(\mathbb{R}^d, \gamma)$. We obtain the exact asymptotic order of
pseudo $s$-numbers for the cases $1 \leq q< p < \infty$ and $p=q=2$.
Additionally, we also obtain an upper bound and a lower bound for pseudo
$s$-numbers of the embedding of $W^\alpha_2(\mathbb{R}^d, \gamma)$ into
$L_{\infty}^{\sqrt{g}}(\mathbb{R}^d)$. Our result is an extension of that
obtained in Dinh D\~ung and Van Kien Nguyen (IMA Journal of Numerical Analysis,
2023) for approximation and Kolmogorov numbers.
