Remarks on counting negative eigenvalues of Schr\"odinger operator on regular metric trees

TL;DR

This paper investigates the number of negative eigenvalues of Schrödinger operators on regular metric trees, establishing conditions under which this number scales with the potential and parameter, and improving upon recent estimates.

Contribution

It provides necessary and sufficient conditions for the eigenvalue count's asymptotic behavior on regular metric trees, extending previous results with sharper estimates.

Findings

Conditions for $N_-(\alpha)$ to be $O(\alpha^p)$ for $p extgreater 1/2$

Necessity and sufficiency of these conditions for special tree classes

Improved estimates over recent literature

Abstract

We discuss estimates on the number $N_-(\alpha)$ of negative eigenvalues of the Schr\"odinger operator $-\Delta-\alpha V$ on regular metric trees, as depending on the properties of the potential $V\ge 0$ and on the value of the large parameter $\alpha$. We obtain conditions on $V$ guaranteeing the behavior $N_-(\alpha)=O(\alpha^p)$ 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 Kova\v{r}\'{i}k, arXive:0710.5500.