Asymptotics of the eigenvalues of the Schrödinger operator

We examine the selfadjoint operator $H=-\Delta+V$ in $L_2(\mathbf R^m)$. We assume that the potential $V(x)\geqslant1$ tends to $+\infty$ as $|x|\to\infty$. Under these conditions the spectrum of $H$ is discrete. In the paper the well-known asymptotic formula
\begin{equation} N(\lambda,H)\sim\gamma_m\int(\lambda-V(x))_+^{m/2}\,dx,\qquad\lambda\to\infty, \tag{\ast} \end{equation}
for the distribution function of the eigenvalues is justified under very weak assumptions on $V$, namely the following conditions:
1) $\sigma(2\lambda)\leqslant c\sigma(\lambda)$, where $\sigma(\lambda)=\operatorname{mes}\{x:V(x)<\lambda\}$;
2) $V(x)\leqslant cV(y)$ almost everywhere when $|x-y|<1$;
3) there exist a continuous function $\eta(t)\geqslant0$, $0\leqslant t<1$, $\eta(0)=0$, and an index $\beta\in[0,1/2)$ such that
$$ \int_{|x-y|\leqslant1,\,|x+z-y|\leqslant1}|V(x+z)-V(x)|\,dx<\eta(|z|)|z|^{2\beta}V(y)^{1+\beta} $$
for any $y\in\mathbf R^m$, $z\in\mathbf R^m$, $|z|<1$.
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