Eigenvalue asymptotics of Schrödinger operators with only discrete spectrum
For the Schrödinger operator with only discrete spectrum \[ - \Delta + V,\;V \in L^ \infty_{\text{loc}} (\mathbb{R}^ n), \quad V(x) \to \infty \quad \text{as} \quad | x | \to \infty \] the author gives new criteria on the potential \(V\) for the applicability of an asymptotic formula for eigenvalues \[ N(\lambda) \sim (2\pi)^{-n} \omega_ n \int_{\mathbb{R}^ n} \bigl( \lambda - V(x) \bigr)_ +^{n/2} dx,\;\lambda \to \infty \] where \(\omega_ n\) is the volume of the unit ball in \(\mathbb{R}^ n\). The author obtains these criteria by means of the Dirichlet-Neumann bracketing method and a modification of some results by Fleckinger and Lapidus (1987). Using these criteria the author proves, for example, the asymptotic formula mentioned above for very slowly growing potentials (such as \(V(x) = \log \dots \log | x |\), \(| x | \to \infty)\) and for nonclassical potentials whose zero sets \(\{x \in \mathbb{R}^ n : V(x) = 0\}\) are not cones (such as \(V(x,y) = \prod^ p_{i = 1} | x - a_ i |^{\alpha_ i} \prod^ q_{j = 1} | y - b_ j |^{\beta_ j}\), \((x,y) \in \mathbb{R}^ 2)\).
scientific article; zbMATH DE number 597464
scientific article