The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities

@article{Frank2021TheNS,
  title={The Nonlinear Schr{\"o}dinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities},
  author={Rupert L. Frank and David Gontier and Mathieu Lewin},
  journal={Communications in Mathematical Physics},
  year={2021},
  volume={384},
  pages={1783-1828}
}
In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator $$-\Delta +V(x)$$ - Δ + V ( x ) are raised to the power $$\kappa $$ κ is never given by the one-bound state case when $$\kappa >\max (0,2-d/2)$$ κ > max ( 0 , 2 - d / 2 ) in space dimension $$d\ge 1$$ d ≥ 1 . When in addition $$\kappa \ge… 
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