Journal of Mathematical Sciences

, Volume 176, Issue 3, pp 458-474

First online:

Spectral estimates for Schrödinger operators with sparse potentials on graphs

  • G. RozenblumAffiliated withChalmers University of Technology, University of Gothenburg Email author 
  • , M. SolomyakAffiliated withThe Weizmann Institute of Science

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A construction of “sparse potentials,” suggested by the authors for the lattice \( {\mathbb{Z}^d} \), d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schrödinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice \( {\mathbb{Z}^2} \), where D = 2. Bibliography: 13 titles.