Spectral Inequalities for Jacobi Operators and Related Sharp Lieb–Thirring Inequalities on the Continuum

Abstract

Abstract. In this paper we approximate a Schrödinger operator on L 2 (R) by Jacobi operators on ℓ 2 (Z) to provide new proofs of sharp Lieb-Thirring inequalities for the powers γ = 1 2 and γ = 3 2 . To this end we first investigate spectral inequalities for Jacobi operators. Using the commutation method we present a new, direct proof of a sharp inequality corresponding to a LiebThirring inequality for the power 3 2 on ℓ 2 (Z). We also introduce inequalities for higher powers of the eigenvalues as well as for matrix-valued potentials and compare our results to previously established bounds. Consider the Schrödinger operatorHere and below, a ± = (|a| ± a)/2 denotes the positive/negative part of a variable, a function or a self-adjoint operator. Subsequently (1) 2 ) and a Weyl-type asymptotic result for the left-hand-side of (1) was also shown to hold in the case γ ≥ 3 2 , d ≥ 1 by Laptev and Weidl [22]. The authors first established Buslaev-Faddeev-Zakharov trace formulae in the more general case of H being defined on L 2 (R, C m ), where the potential is a Hermitian matrix-valued function V : R → C m×m . This yielded sharp Lieb-Thirring inequalities in one dimension for matrix-valued potentials and γ ≥ 32 . An induction argument making use of the matrix nature of the results lifted the inequalities to higher dimensions.For γ = 1 2 , d = 1 Hundertmark, Lieb and Thomas [18] proved that the best constant is given by L 1/2,1 = 2L cl 1/2,1 = 1 2 . The inequality is then sharp for delta potentials. In [12] it was shown that for 1 ≤ γ < 3 2 the sharp constant can be bounded by L γ,1 ≤ π √ 3 L cl γ,1 , which was generalised to higher dimensions in [11].In our paper we provide a new proof for sharp Lieb-Thirring bounds in one dimension for γ = 1 2 and γ = 3 2 using an approximation of the Schrödinger operator by Jacobi operators on the lattice.

The state of the Lieb--Thirring conjecture

“…In the former case, the monotonicity and ensuing bound had also been proved earlier by Hundertmark and Simon [32] following the arguments of [34]. Using this discrete result together with an approximation argument [62] yields an alternative proof of Theorem 9. Helffer and Robert [28] constructed examples that show…”

“…For the latter operators, corresponding trace formulae were established by Deift and Killip [10] as well as Killip and Simon [38]. Using an approximation argument [62], the Lieb-Thirring bound in the discrete setting can be used to provide an alternative proof of R 3/2,1 = 1. The Lieb-Thirring inequality (2) in the endpoint case d = 1, γ = 1/2 was first proved by Weidl [64], 20 years after the original paper by Lieb and Thirring.…”

“…The argument relies on the so-called commutation method to successively remove eigenvalues from the spectrum of the operator. This method has also been applied to establish sharp Lieb-Thirring type inequalities for Schrödinger operators on the half-axis with Robin boundary condition [13], for fourth order operators [29] and for discrete Jacobi operators [62]. For the latter operators, corresponding trace formulae were established by Deift and Killip [10] as well as Killip and Simon [38].…”

The Lieb–Thirring inequalities: Recent results and open problems

“…Lieb-Thirring inequalities for discrete Schrödinger operators. Results for Jacobi matrices and discrete Schrödinger operators can be found, for instance, in [95,100,164,169,170,4] and in the references therein. Due to the lack of scaling invariance the form of the inequality and therefore also the question of optimal constants is less clear in this setting.…”

Improved sharp spectral inequalities for Schrödinger operators on the semi-axis

“…Subsequently, it has been applied to provide a new, direct proof of (1) in the case of matrix-valued potentials [1] (as first established by Laptev and Weidl [12]) and to prove similar inequalities for fourth-order differential operators [10] and Jacobi operators [17]. In a slight variation, this proof method has also been used to establish Theorem 3.…”