Remarks on counting negative eigenvalues of Schrödinger operator on regular metric trees
| dc.creator | Solomyak, Michael | |
| dc.date | 2008-09-04 | |
| dc.date.accessioned | 2026-07-07T10:00:39Z | |
| dc.date.available | 2026-07-07T10:00:39Z | |
| dc.description | We discuss estimates on the number $N_-(α)$ of negative eigenvalues of the Schrödinger operator $-Δ-αV$ on regular metric trees, as depending on the properties of the potential $V\ge 0$ and on the value of the large parameter $α$. We obtain conditions on $V$ guaranteeing the behavior $N_-(α)=O(α^p)$ for any given $p\ge 1/2$. For a special class of trees we show that these conditions are not only sufficient but also necessary. For $p>1/2$ the order-sharp estimates involve a (quasi-)norm of $V$ in some `weak' $L_p$- or $\ell_p(L_1)$-space. We show that the results can be easily derived from the ones of an earlier paper by Naimark and the author, Proc. London Math. Soc. (3) 80 (2000), 690-724. The results considerably improve the estimates found in the recent paper by Ekholm, Frank, and Kovařík, arXive:0710.5500. | |
| dc.description | 19 pages | |
| dc.identifier | https://arxiv.org/abs/0809.0752 | |
| dc.identifier | http://arxiv.org/abs/0809.0752 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/168378 | |
| dc.subject | Spectral Theory | |
| dc.subject | Mathematical Physics | |
| dc.title | Remarks on counting negative eigenvalues of Schrödinger operator on regular metric trees | |
| dc.type | text |