Eigenvalue estimates for singular left-definite Sturm–Liouville operators

Behrndt, Jussi; Möws, Roland; Trunk, Carsten

doi:10.4171/jst/14

Cited by 10 publications

(10 citation statements)

References 25 publications

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“…The main result Theorem 4.1 extends the estimate in [12,Theorem 4.1]. In contrast to [12] we go beyond the so-called left-definite case, which was studied intensively from different points of view; cf. [16][17][18]20,38,40,41,56].…”

Section: Singular Indefinite Sturm-liouville Problemsmentioning

confidence: 87%

“…In Section 4 we show how our general eigenvalue estimates can be applied to indefinite singular Sturm-Liouville problems. We consider the situation where the associated operator is nonnegative in an L 2 -Krein space and, in this specific situation, the estimates from Section 3 can be slightly improved and lead to a generalization of [12,Theorem 4.1]. In particular, this also includes the so-called left definite Sturm-Liouville problems where the associated operator is uniformly positive in an L 2 -Krein space; cf.…”

Section: Introductionmentioning

confidence: 96%

“…In particular, this also includes the so-called left definite Sturm-Liouville problems where the associated operator is uniformly positive in an L 2 -Krein space; cf. [12,[16][17][18]20,38,40,41,56] for related work on left definite problems.…”

Section: Introductionmentioning

confidence: 99%

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Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces

Behrndt

Leben

Peria

et al. 2016

Journal of Mathematical Analysis and Applications

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Let A and B be selfadjoint operators in a Krein space and assume that the resolvent difference of A and B is of rank one. In the case that A is nonnegative and I is an open interval such that σ(A) ∩ I consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of B in I. The general result is illustrated with eigenvalue estimates for singular indefinite Sturm-Liouville problems.

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Section: Singular Indefinite Sturm-liouville Problemsmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 96%

See 1 more Smart Citation

Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces

Behrndt

Leben

Peria

et al. 2016

Journal of Mathematical Analysis and Applications

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“…in (2.1), but now the extension T 0 of T 0 is defined on the entire Sobolev scale according to (3.11), 14) and the special case s = 1 again corresponds to (C.40),…”

Section: One Obtains the Scale Of Sobolev Spaces Viamentioning

confidence: 99%

Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

Gesztesy

Weikard

2014

Operator Theory: Advances and Applications

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Dedicated with great pleasure to Ludwig Streit on the occasion of his 75th birthday.Abstract. We study particular cases of left-definite eigenvalue problems Aψ = λBψ, with A ≥ εI for some ε > 0 and B self-adjoint, but B not necessarily positive or negative definite, applicable, in particular, to the eigenvalue problem underlying the Camassa-Holm hierarchy. In fact, we will treat a more general version where A represents a positive definite Schrödinger or Sturm-Liouville operator T in L 2 (R; dx) associated with a differential expression of the formrepresents an operator of multiplication by r(x) in L 2 (R; dx), which, in general, is not a weight, that is, it is not nonnegative (or nonpositive) a.e. on R. In fact, our methods naturally permit us to treat certain classes of distributions (resp., measures) for the coefficients q and r and hence considerably extend the scope of this (generalized) eigenvalue problem, without having to change the underlying Hilbert space L 2 (R; dx). Our approach relies on rewriting the eigenvalue problem Aψ = λBψ in the formand a careful study of (appropriate realizations of) the operator A −1/2 BA −1/2 in L 2 (R; dx).In the course of our treatment, we review and employ various necessary and sufficient conditions for q to be relatively bounded (resp., compact) and relatively form bounded (resp., form compact) with respect to T 0 = −d 2 /dx 2 defined on H 2 (R). In addition, we employ a supersymmetric formalism which permits us to factor the second-order operator T into a product of two firstorder operators familiar from (and inspired by) Miura's transformation linking the KdV and mKdV hierarchy of nonlinear evolution equations. We also treat the case of periodic coefficients q and r, where q may be a distribution and r generates a measure and hence no smoothness is assumed for q and r.

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“…[15,21,22,25,31,34,38,39,41,66,93], KreinFeller operators [46], λ-dependent boundary value problems, see e.g. [24,30,42,62,63,82], operator polynomials [68,69,71,72,74,75,76,87], second order systems [50,89,90] and in the study of problems of Klein-Gordon type [83].…”

Section: Definition 24 [7] For a Selfadjoint Operatormentioning

confidence: 99%