Article

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

10/2008;
Source: arXiv

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.

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• Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, in: Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad. M Birman, Sh, M Solomyak . 1989. Adv. Soviet Math 7 1-55.
• Schrödinger operator Estimates for number of bound states as function-theoretical problem, in: Spectral theory of operators (Novgorod. M Birman, Sh, M Z Solomyak . 1989. Amer. Math. Soc. Transl. Ser 2 1-54.
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### Keywords

estimates

large parameter $\alpha$

negative eigenvalues

order-sharp estimates

potential $V\ge 0$

quasi-)norm

regular metric trees

Schr\"odinger operator $-\Delta-\alpha V$

Soc

special class

trees