Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities

Lieb, Élliott H.; Thirring, Walter

doi:10.1007/3-540-27056-6_16

Cited by 287 publications

(415 citation statements)

References 11 publications

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“…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…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2006

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“…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…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2006

Self Cite

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“…The following theorem gives an improvement of the result in [1], where we treated only the case for the first eigenvalue λ 1 by use of a Lieb-Thirring inequality [8].…”

Section: Theoremmentioning

confidence: 99%

On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator

Ichinose

Wakayama²

2007

Reports on Mathematical Physics

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The spectral zeta function for the so-called non-commutative harmonic oscillator is able to be meromorphically extended to the whole complex plane, having only one simple pole at the same point s = 1 where Riemann's zeta function ζ(s) has, and possesses a trivial zero at each non-positive even integer. The essential part of its proof is sketched. A new result is also given on the lower and upper bounds of the eigenvalues of the non-commutative harmonic oscillator.

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“…Surprisingly, many evolution equations generate finite-dimensional attractors. The remarkable Sobolev-LiebThirring inequalities [19,32] have been used to establish physically relevant upper bounds on the box-counting and Hausdorff dimensions of the global attractors of many evolution equations. Examples include nonlinear wave equations, reactiondiffusion equations, and the two-dimensional incompressible Navier-Stokes system.…”

Section: That S(t)a = a For All T ∈ R And Dist(s(t)d A) → 0 As T → ∞mentioning

confidence: 99%

“…Such attractors exist for a variety of the evolution equations of mathematical physics, including the Navier-Stokes system, various classes of reactiondiffusion systems, nonlinear dissipative wave equations, and complex GinzburgLandau equations. The remarkable Sobolev-Lieb-Thirring inequalities [32,19] have been invoked to establish physically significant upper bounds on attractor dimension in a number of cases. Nevertheless, a fundamental question remains.…”

Section: Conjecture 84 ([43]mentioning

confidence: 99%

Untitled

2005

Bull. Amer. Math. Soc.

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Abstract. Many problems in mathematics and science require the use of infinite-dimensional spaces. Consequently, there is need for an analogue of the finite-dimensional notions of 'Lebesgue almost every' and 'Lebesgue measure zero' in the infinite-dimensional setting. The theory of prevalence addresses this need and provides a powerful framework for describing generic behavior in a probabilistic way. We survey the theory and applications of prevalence.

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Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities

Cited by 287 publications

References 11 publications

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

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

On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator

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