Asymptotics of the eigenvalues of the Schr\"odinger operator

G. V. Rozenblum

Sbornik. Mathematics, Tome 22 (1974) no. 3, pp. 349-371

Résumé

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 $\eta(t)\geqslant0$, $0\leqslant t1$, $\eta(0)=0$, 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.