Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory

Бирман, М. Ш.; Solomjak, M. Z.

doi:10.1090/trans2/114

Cited by 113 publications

(116 citation statements)

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“…The inequality (3.17a) is a triangle inequality for the functional ∆ p p+1 p,ϕ . The inequalities (3.17a), (3.17b) strengthen the well-known Weyl's theorem ( [We], see also [BS1], Lemma 1.17). They mean that the functionals ∆ p,ϕ and δ p,ϕ are continuous in a certain sense.…”

Section: Ifsupporting

confidence: 72%

“…They mean that the functionals ∆ p,ϕ and δ p,ϕ are continuous in a certain sense. These inequalities are well-known for ϕ = 1 (see [RSSh], Subsection 11.1; the proofs can be found also in [BS1], Lemma 1.18). The general case was treated in [BS3], Lemma 3.1.…”

Section: Ifmentioning

confidence: 91%

“…Next, due to [BS1], Appendix 7, the marginal integral operators under consideration satisfy the conditions of Theorem 8.1. Then we can put Ψ = √ ρ, and the statement follows.…”

Section: Some Generalizationsmentioning

confidence: 99%

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Small ball probabilities for Gaussian random fields and tensor products of compact operators

Karol

Nazarov

Nikitin

2008

Trans. Amer. Math. Soc.

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Abstract. We find the logarithmic L 2 -small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein -Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csáki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.

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Section: Ifsupporting

confidence: 72%

Section: Ifmentioning

confidence: 91%

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Small ball probabilities for Gaussian random fields and tensor products of compact operators

Karol

Nazarov

Nikitin

2008

Trans. Amer. Math. Soc.

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“…In the case of Dirichlet boundary conditions these asymptotics go back to [15]. They have been generalized in various ways, in particular, to the case of Robin boundary conditions (1.2); see, for instance, the lecture notes [2]. It has been conjectured by Weyl that (1.3) is the beginning of an asymptotic expansion in n and that the second term should depend on the surface area of .…”

Section: Introductionmentioning

confidence: 96%

Semi-classical analysis of the Laplace operator with Robin boundary conditions

Frank

Geisinger

2012

Bull. Math. Sci.

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We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in terms of semi-classical analysis. In this reformulation it is natural to allow the function describing the boundary conditions to depend on the semi-classical parameter and we identify and analyze three different regimes for this dependence.

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“…A simple sufficient condition for such behavior was obtained as far back as in 1974, see [4], §4.8. The direct analogue of (1.3) is valid only for the monotone potentials V (Calogero estimate, see e.g.…”

Section: 1)mentioning

confidence: 95%

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

Лаптев

Solomyak

2012

Commun. Math. Phys.

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Abstract. For a two-dimensional Schrödinger operator H αV = −∆ − αV with the radial potential V (x) = F (|x|), F (r) ≥ 0, we study the behavior of the number N − (H αV ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N − (H αV ) = O(α) and for the validity of the Weyl asymptotic law.

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Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory

Cited by 113 publications

References 0 publications

Small ball probabilities for Gaussian random fields and tensor products of compact operators

Small ball probabilities for Gaussian random fields and tensor products of compact operators

Semi-classical analysis of the Laplace operator with Robin boundary conditions

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

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