Some properties of block-radial functions and Schr\"odinger type operators with block-radial potentials

TL;DR

This paper investigates the entropy numbers of embeddings of block-radial Besov spaces and applies these results to estimate the negative spectra of Schrödinger operators with block-radial potentials, revealing dependence on block structure.

Contribution

It provides new asymptotic estimates for entropy numbers of block-radial Besov space embeddings and links these to spectral properties of Schrödinger operators with block-radial potentials.

Findings

Entropy numbers depend on the number of blocks of lowest dimension and parameters p1, p2.

Asymptotic behaviour is independent of smoothness parameters s1, s2.

Application to spectral estimates of Schrödinger operators with block-radial potentials.

Abstract

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.