On gaps between bounded operators

Key words Elliptic operators, Dirichlet boundary conditions, stability estimates for the eigenfunctions, perturbation of an open set, gap between linear operators MSC (2010) 47F05, 35J40, 35B30, 35P15 Stability of the eigenfunctions of nonnegative selfadjoint second-order linear elliptic operators subject to homogeneous Dirichlet boundary data under domain perturbation is investigated. Let Ω, Ω ⊂ R n be bounded open sets. The main result gives estimates for the variation of the eigenfunctions under perturbations Ω of Ω such that Ωε = {x ∈ Ω : dist(x, R n \Ω) > ε} ⊂ Ω ⊂ Ω ⊂ Ω in terms of powers of ε, where the parameter ε > 0 is sufficiently small. The estimates obtained here hold under some regularity assumptions on Ω, Ω . They are obtained by using the notion of a gap between linear operators, which has been recently extended by the authors to differential operators defined on different open sets.

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“…In virtue of Theorem 2.5 it follows also that δ(0, T ) = δ(T, 0) =δ(T, 0). Further results with regards to bounded operator can be found in [6].…”

Section: Gap Between Operatorsmentioning

confidence: 99%

“…(1. 6) then lim k →∞ λ k [Ω] = ∞. Moreover, there exists an orthonormal basis of eigenfunctions {ϕ k [Ω]} k ∈N in L 2 (Ω).…”

Section: Introductionmentioning

confidence: 99%

Spectral stability estimates for the eigenfunctions of second order elliptic operators

Burenkov

Feleqi

2012

Mathematische Nachrichten

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“…(v) In the characterization of non-reversing random walks [30] on G, though here the stellated graph has only the secondary external arc set E * = {i j j k : i j , j k ∈ V * , i = k}. (vi) For the case of a regular graph (which is stellated) in a work by Cvetkovic [31], without naming the graphs-see theorem 6. (vii) Yet further they arise [32] in describing rovibronic eigenspectra arising from different minima in degenerate molecular rearrangements.…”

Section: Introductionmentioning

confidence: 99%

Two-point resistances and random walks on stellated regular graphs

Yang

Klein

2019

J. Phys. A: Math. Theor.

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A formula for computing the resistance between any two vertices in the stellated graph of a regular graph is obtained. It turns out that the two-point resistance of the stellated graph can be expressed in terms of the two-point resistance of the original graph. As a consequence, the Kirchhoff index (i.e. the sum of the effective resistances between all pairs of vertices) for the stellated graph is obtained, which extends the previously known result. The correspondence between random walks and electric networks is then used to obtain the mean first passage time and mean commute time for random walks on stellated graphs.

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“…Some relations among the gap and the spherical gap of a bounded operator on Banach spaces were studied by Cvetković in [4].…”

Section: Introductionmentioning

confidence: 99%

The Horn-Li-Merino formula for the gap and the spherical gap of unbounded operators

Ramesh

2011

Proc. Amer. Math. Soc.

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Abstract. In this article we obtain the Horn-Li-Merino formula for computing the gap as well as the spherical gap between two densely defined unbounded closed operators. As a consequence we prove that the gap and the spherical gap of an unbounded closed operator are 1 and √ 2 respectively. With the help of these formulae we establish a relation between the spherical gap and the gap of unbounded closed operators. We discuss some properties of the spherical gap similar to those of the gap metric.

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On gaps between bounded operators

Cited by 20 publications

References 2 publications

Spectral stability estimates for the eigenfunctions of second order elliptic operators

Spectral stability estimates for the eigenfunctions of second order elliptic operators

Two-point resistances and random walks on stellated regular graphs

The Horn-Li-Merino formula for the gap and the spherical gap of unbounded operators

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