An asymptotic of the negative discrete spectrum of the Schr\"odinger operator

G. V. Rozenblum

Matematičeskie zametki, Tome 21 (1977) no. 3, pp. 399-407.

Résumé

The Schrödinger operator $Hu=-\Delta u+V(x)u$, where $V(x)\to0$ as $|x|\to\infty$, is considered in $L_2(R^m)$ for $m\ge3$. The asymptotic formula $$ N(\lambda,V)\sim\gamma_m\int(\lambda-V(x))^{m/2}_+\,dx\quad\lambda\to-0. $$ is established for the number $N(\lambda,V)$ of the characteristic values of the operator $H$ which are less than $\lambda$. It is assumed about the potential $V$ that $V=V_0+V_1$; $V_00$, $|\nabla V_0|=o(|V_0|^{3/2})$ as $|x|\to\infty$; $\sigma(t/2,V_0)\le c\sigma(t,V_0)$ and $V_1\in L_{m/2,\operatorname{loc}}$, $\sigma(t,V_1)=o(\sigma(t,V_0))$, where $\sigma(t,f)=\operatorname{mes}\{x:|f(x)|>t\}$.