On Riesz Means of Eigenvalues

Harrell, Evans M.; Hermi, Lotfi

doi:10.1080/03605302.2011.595865

Cited by 22 publications

(16 citation statements)

References 44 publications

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“…Since we focused on the case of nonconstant coefficients throughout, we did not enter the vast literature on eigenvalue counting function estimates in connection with the Laplacian and its (fractional) powers. In this context we refer, for instance, to [34], [48], [97], [98], and the extensive literature cited therein. ⋄…”

Section: An Upper Bound For the Eigenvalue Counting Function For The mentioning

confidence: 99%

“…We emphasize that (1.17) is not in conflict with variational eigenvalue estimates since N (λ; A K,Ω,2m (a, b, q)) only counts the strictly positive eigenvalues of A K,Ω,2m (a, b, q) less than λ > 0 and hence avoids taking into account the (generally, infinite-dimensional) null space of A K,Ω,2m (a, b, q). Rather than relying on estimates for N ( · ; A F,Ω,2m (a, b, q)) (cf., e.g., [12]- [18], [36], [37], [47], [48], [61], [63], [64], [68], [71], [78], [79], [81], [83], [95], typically for a = I n , b = 0), we will use the one-to-one correspondence of nonzero eigenvalues of A K,Ω,2m (a, b, q) with the eigenvalues of its underlying buckling problem (cf. (1.3)-(1.5)) and estimate the eigenvalue counting function for the latter.…”

Section: Introductionmentioning

confidence: 99%

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A bound for the eigenvalue counting function for Krein–von Neumann and Friedrichs extensions

Ashbaugh

Gesztesy

Лаптев

et al. 2017

Advances in Mathematics

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Dedicated with great pleasure to Yuri Latushkin on the occasion of his 60th birthday. Abstract.For an arbitrary open, nonempty, bounded set Ω ⊂ R n , n ∈ N, and sufficiently smooth coefficients a, b, q, we consider the closed, strictly positive, higher-order differential operator A Ω,2m (a, b, q) in L 2 (Ω) defined on W 2m,2 0 (Ω), associated with the higher-order differential expressionand its Krein-von Neumann extension A K,Ω,2m (a, b, q) in L 2 (Ω). Denoting by N (λ; A K,Ω,2m (a, b, q)), λ > 0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of A K,Ω,2m (a, b, q), we derive the boundwhere C = C(a, b, q, Ω) > 0 (with C(In, 0, 0, Ω) = |Ω|) is connected to the eigenfunction expansion of the self-adjoint operator A 2m (a, b, q) in L 2 (R n ) defined on W 2m,2 (R n ), corresponding to τ 2m (a, b, q). Here vn := π n/2 /Γ((n + 2)/2) denotes the (Euclidean) volume of the unit ball in R n .Our method of proof relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of A 2 (a, b, q) in L 2 (R n ).We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension A F,Ω,2m (a, b, q) in L 2 (Ω) of A Ω,2m (a, b, q).No assumptions on the boundary ∂Ω of Ω are made. Date: July 4, 2018. 2010 Mathematics Subject Classification. Primary 35J25, 35J40, 35P15; Secondary 35P05, 46E35, 47A10, 47F05.Key words and phrases. Krein and Friedrichs extensions of general second-order uniformly elliptic partial differential operators, bounds on eigenvalue counting functions, spectral analysis, buckling problem.

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Section: An Upper Bound For the Eigenvalue Counting Function For The mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A bound for the eigenvalue counting function for Krein–von Neumann and Friedrichs extensions

Ashbaugh

Gesztesy

Лаптев

et al. 2017

Advances in Mathematics

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“…Reversely, to recover sharp Berezin-Li-Yau bounds from Kac' inequality one needs some additional information. For example, in [HH07] Harrell and Hermi formally deduced Berezin-Li-Yau bounds for σ ≥ 2 from Kac' inequality based on a monotonicity result by Harrell and Stubbe. 1 Similar arguments fail for σ < 2.…”

Section: Z(t) − |ω| (4πt)mentioning

confidence: 99%

“…Moreover, one can use the ideas of [Mel03] and [HH07] to derive unviersal bounds on Z(t). We can employ inequality (5) and the result of Luttinger (7).…”

Section: Proof Of Theorem 1 and Remarksmentioning

confidence: 99%

Universal bounds for traces of the Dirichlet Laplace operator

Geisinger¹,

Weidl²

2010

Journal of the London Mathematical Society

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Abstract. We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set Ω ⊂ R d , d ≥ 2. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on Z(t) in domains of infinite volume.For domains of finite volume the bound on Z(t) decays exponentially as t tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of Z(t). To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means.

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“…The Engel group is a Carnot group of step G 3 r  (see [1]), its Lie algebra is generated by the left-invariant vector fields The Riesz means of Dirichlet eigenvalues for the Laplace operator in the Euclidean space have been extensively studied(see [3][4][5]). In recent years, E. M. Harrell II and L. Hermi in [6] treated the Riesz means…”

Section: Introductionmentioning

confidence: 99%