On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

Ouhabaz, El Maati

doi:10.1007/pl00005980

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Cited by 2 publications

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“…The operator H is then defined as the self-adjoint operator associated with its closure. There are various sufficient conditions for the closability of Q, for which we refer to [3,5,8,9] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Trace estimates and invariance of the essential spectrum

Barbatis

2006

Arch. Math.

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We provide sufficient conditions under which the difference of the resolvents of two higher-order operators acting in R N belongs to trace classes C p . We provide explicit estimates on the norm of the resolvent difference in terms of L p norms of the difference of the coefficients. Such inequalities are useful in estimating the effect of localized perturbations of the coefficients. AMS 2000 MSC: 35P05 (35J30, 47F05)

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Section: Introductionmentioning

confidence: 99%

Trace estimates and invariance of the essential spectrum

Barbatis

2006

Arch. Math.

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The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds

Ouhabaz¹

2001

Duke Math. J.

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Given a complete Riemannian manifold M and a Schrödinger operator − + m acting on L p (M), we study two related problems on the spectrum of − +m. The first one concerns the positivity of the L 2 -spectral lower bound s(− + m). We prove that if M satisfies L 2 -Poincaré inequalities and a local doubling property, then s(− + m) > 0, provided that m satisfies the mean condition inf p∈Mfor some r > 0. We also show that this condition is necessary under some additional geometrical assumptions on M.The second problem concerns the existence of an L p -principal eigenvalue, that is, a constant λ ≥ 0 such that the eigenvalue problem u = λmu has a positive solution u ∈ L p (M). We give conditions in terms of the growth of the potential m and the geometry of the manifold M which imply the existence of L p -principal eigenvalues.Finally, we show other results in the cases of recurrent and compact manifolds.

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On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

Cited by 2 publications

References 9 publications

Trace estimates and invariance of the essential spectrum

Trace estimates and invariance of the essential spectrum

The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds

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