Sharp Lieb-Thirring inequalities in high dimensions

Лаптев, Ари; Weidl, Timo

doi:10.1007/bf02392782

Cited by 170 publications

(216 citation statements)

References 28 publications

Supporting

Mentioning

210

Contrasting

Unclassified

Order By: Relevance

“…Lieb-Thirring inequalities for matrix-valued potentials for the value γ = 3/2 were obtained in [6] and also in [2]. Here we state a result corresponding to γ = 1.…”

Section: Introductionmentioning

confidence: 70%

See 1 more Smart Citation

Lieb–Thirring inequalities with improved constants

Dolbeault¹,

Лаптев²,

Loss³

2008

J. Eur. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Lieb-Thirring inequalities for matrix-valued potentials for the value γ = 3/2 were obtained in [6] and also in [2]. Here we state a result corresponding to γ = 1.…”

Section: Introductionmentioning

confidence: 70%

“…By using the Aizenman-Lieb argument [1], a "lifting" with respect to dimension [6], [5], and Theorem 1 we obtain the following result:…”

Section: Remarkmentioning

confidence: 95%

Lieb–Thirring inequalities with improved constants

Dolbeault¹,

Лаптев²,

Loss³

2008

J. Eur. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Here, as usual, the double braces denote the integer part. This lower limit has the merit of being rather neat, but it increases proportionally to g 2 (see (13)) rather than g 3 (see (16)), hence it cannot be expected to be cogent for strong potentials possessing many bound states.…”

Section: Limits On the Total Number Of Bound Statesmentioning

confidence: 99%

Upper and lower limits on the number of bound states in a central potential

Brau

Calogero²

2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

In a recent paper new upper and lower limits were given, in the context of the Schrödinger or Klein-Gordon equations, for the number N 0 of S-wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity. In this paper these results are extended to the number N of bound states for the th partial wave, and results are also obtained for potentials that are not monotonic and even somewhere positive. New results are also obtained for the case treated previously, including the remarkably neat lower limit N {{[σ/(2 +1)+1]/2}} with σ = (2/π ) max 0 r<∞ [r|V (r)| 1/2 ] (valid in the Schrödinger case, for a class of potentials that includes the monotonically nondecreasing ones), entailing the following lower limit for the total number N of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity: N {{(σ +1)/2}}{{(σ +3)/2}}/2 (here the double braces denote the integer part).

show abstract

“…One can reduce a multidimensional problem to a one dimensional matrixvalued one using the main idea of Laptev and Weidl [18]. The result can be formulated as follows.…”

Section: The Laptev-weidl Methodsmentioning

confidence: 99%

Cwikel and Quasi-Szego Type Estimates for Random Operators

Holt

Molchanov

Safronov

2008

LPDE

View full text Add to dashboard Cite

ABSTRACT. We consider Schrodinger operators with nonergodic random potentials. Specifically, we are interested in eigenvalue estimates and estimates of the entropy for the absolutely continuous part of the spectral measure. We prove that increasing oscillations in the potential at infinity have the same effect on the properties of the spectrum as the decay of the potential.Recall the Rozenbljum-Cwikel-Lieb estimate ([5], [20], [19], [27]) for the number N (V ) of negative eigenvalues of −∆ − V (x):The potential V in this estimate must decay in order to make the integral converge. We are going to study potentials that either decay slower than L d/2 -potentials or do not decay at all. Instead of decay, our theorems require some oscillation of V at infinity. There is an additional disadvantage of our results in that they give an estimate for the number of eigenvalues below arbitrary negative number −γ, which can not be taken equal to zero. In order to estimate the amount of negative spectrum one has to combine our main result with the Laptev-Weidl approach (see Theorem (1.2)). We also study the conditions on V which guarantee the convergence of certain eigenvalue sums. The classical Lieb-Thirring estimate for the eigenvalue sum |λ j | γ holds for all potentials from L d/2+γ , even for the worst ones. Our goal is to show that the probability to meet a "bad" potential is zero. It means that for a typical potential one can expect a better behaviour of the negative spectrum.Denote by F the Fourier transform understood as a unitary operator inwhere a is the multiplication operator by a function denoted by the same letter and V is the multiplication operator by a potential V . First, we need to state the Birman-Schwinger principle, which can be formulated as follows:Proposition 0.1. Let H = −∆ − V be the Schrödinger operator with a real potential V and let a(ξ) = (ξ 2 + γ) −1/2 . Then the number of eigenvalues 1

show abstract

Sharp Lieb-Thirring inequalities in high dimensions

Cited by 170 publications

References 28 publications

Lieb–Thirring inequalities with improved constants

Lieb–Thirring inequalities with improved constants

Upper and lower limits on the number of bound states in a central potential

Cwikel and Quasi-Szego Type Estimates for Random Operators

Contact Info

Product

Resources

About