Compactness of multiplication operators

Question

Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the compactness of the embeddding from $H^1(B)$ to $L^q(B)$, $2 \le q <2^*$, when $B$ is a ball. If $p=n/2$ one need also an approximation argument. I am interested in the approximation numbers of $T$ $$a_n(T)=\inf \{\|T-S\|: {\rm rank\ } S <n\}.$$ I think they should be known, at least in the asymptotic behavior. I do not know the answer when $p=n/2$ even in the case where $R^n$ is replaced by a ball.