Recent results on Lieb-Thirring inequalities

Sharp Semiclassical Bounds for the Moments of Eigenvalues for Some Schrödinger Type Operators with Unbounded Potentials

“…In fact, inequality (1.1) is the Legendre transformation of an earlier result proved in [1], as it was pointed out in [10]. In the present article we show that the method developed in [7] can be used to establish the sharp upper bound for the moments of the Dirichlet eigenvalues λ 0 k of certain negative powers.…”

Number of Bound States of Schrödinger Operators with Matrix-Valued Potentials

“…The motivation for this extension was the work of Laptev and Weidl [LW1] who realized that the extension allowed one to conclude that good/sharp constants obtained in low dimensions would automatically give good/sharp constants in higher dimensions. The fact that the inequality (1.1) is valid in the matrix case was proved by Hundertmark [H], confirming a conjecture in [LW2]. He follows Cwikel's method and obtains a constant which is far from optimal.…”

Lieb–Thirring Inequalities for Schrödinger Operators with Complex-valued Potentials

“…Before stating our main results let us recall the standard Lieb-Thirring inequalities (see [5] and also the survey [4]). For real-valued potentials V one has the bound…”

“…(Here and in the sequel t − := max{0, −t} denotes the negative part of t.) By L γ,d we will always mean the sharp constant in (3) (which at present is only known for [4]). For general, complex-valued potentials we shall prove Theorem 1 (Eigenvalue sums).…”