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Asymptotics of the eigenvalues of the Schrödinger operator
G. V. Rozenblum
Abstract:
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
$y\in\mathbf R^m$,
$z\in\mathbf R^m$,
$|z|<1$.
Bibliography: 12 titles.
UDC:
517.43
MSC: 35J10,
35P20,
47F05 Received: 19.01.1973