On Lieb-Thirring Inequalities for Schrödinger Operators with Virtual Level

  title={On Lieb-Thirring Inequalities for Schr{\"o}dinger Operators with Virtual Level},
  author={Tomas Ekholm and Rupert L. Frank},
  journal={Communications in Mathematical Physics},
  • T. Ekholm, R. Frank
  • Published 10 February 2006
  • Mathematics
  • Communications in Mathematical Physics
We consider the operator H=−Δ−V in L2(ℝd), d≥3. For the moments of its negative eigenvalues we prove the estimate Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator. 
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  • On spectral theory of elliptic operators. Oper. Theory Adv. Appl. 89, Birkhäuser, Basel,
  • 1996