On spectral estimates for two-dimensional Schrödinger operators

  • Ari Laptev

    Imperial College London, UK
  • Michael Solomyak

    Weizmann Institute of Science, Rehovot, Israel


For the two-dimensional Schrödinger operator HαV=ΔαV, V0\mathrm H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0, we study the behavior of the number N(HαV)N_-(\mathrm H_{\alpha V}) of its negative eigenvalues (bound states), as the coupling parameter α\alpha tends to infinity. A wide class of potentials is described, for which N(HαV)N_-(\mathrm H_{\alpha V}) has the semi-classical behavior, i.e. N(HαV)=O(α)N_-(\mathrm H_{\alpha V})=O(\alpha). For the potentials from this class, the necessary and sufficient condition is found for the validity of the Weyl asymptotic law.

Cite this article

Ari Laptev, Michael Solomyak, On spectral estimates for two-dimensional Schrödinger operators. J. Spectr. Theory 3 (2013), no. 4, pp. 505–515

DOI 10.4171/JST/53