  Let $ R_\gamma B^{s}_{p,q}(\Rd)$ be a subspace of the Besov space
$B^{s}_{p,q}(\Rd)$ that consists of block-radial functions. We prove that the
asymptotic behaviour of the entropy numbers of compact embeddings $\id: \:
R_\gamma B^{s_1}_{p_1,q_1}(\R^d) \rightarrow R_\gamma B^{s_2}_{p_2,q_2}(\R^d)$
depends on the number of blocks of the lowest dimension, the parameters $p_1$
and $p_2$, but is independent of the smoothness parameters $s_1$, $s_2$. We
apply the asymptotic behaviour to estimation of powers of a negative spectra of
Schr\"odinger type operators with block-radial potentials. This part
essentially relies on the Birman-Schwinger principle.
