Abstract
We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the negative eigenvalues of Schr\odinger operators, which significantly improve the so far existing bounds.
Highlights
In 1975, Lieb and Thirring [19, 20] proved that the sum of all negative eigenvalues of Schrodinger operators −∆ + V in L2(Rd), with a real-valued potential V : Rd → R, admits the bound
It can be stated as a lower bound on the fermionic kinetic energy, Tr(−∆γ) ≥ Kd γ(x, x)1+
Transfering the latter to a kernel bound, using the same computation as in (46)-(47), and optimizing over ε > 0 and E > 0 we obtain the following analogue of (41),
Summary
In 1975, Lieb and Thirring [19, 20] proved that the sum of all negative eigenvalues of Schrodinger operators −∆ + V in L2(Rd), with a real-valued potential V : Rd → R, admits the bound. The Lieb–Thirring inequality was invented to prove the stability of matter [19] In this context, it can be stated as a lower bound on the fermionic kinetic energy, Tr(−∆γ) ≥ Kd γ(x, x)1+. We take into account a low momentum averaging This improves further the bound to L1,1/Lc1l,1 ≤ 1.456 in d = 1 (and worse estimates in higher dimensions). This is one of our key ideas and deviates substantially from Rumin’s original argument. The main ingredients of the proof of Theorem 1, except the lifting argument, apply well to the fractional case New ideas are certainly needed to attack the full conjecture (4)
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