Some properties of block-radial functions and Schrödinger type operators with block-radial potentials

Abstract

Let $ R_γB^{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 $\id: \: R_γB^{s_1}_{p_1,q_1}(\R^d) \rightarrow R_γB^{s_2}_{p_2,q_2}(\R^d)$ 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 Schrödinger type operators with block-radial potentials. This part essentially relies on the Birman-Schwinger principle.

arXiv comment31 pages