Geometric Aspects of Spectral Theory

Berg, M. van den; Grieser, Daniel; Hoffmann‐Ostenhof, Thomas; Polterovich, Iosif

doi:10.4171/owr/2012/33

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…For details see [Fle99,Lemma 6]. Recently, based on [GKMV13], more general operators of the form div(C∇·) with Dirichlet boundary conditions and indefinite Hermitian coefficient matrices C have been investigated [HKKS12]. Again these results do not cover Example 4.16 in the case b = −a.…”

Section: The M C Intosh Conditionmentioning

confidence: 91%

A generalisation of the form method for accretive forms and operators

Elst

Sauter

Vogt

2015

Journal of Functional Analysis

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Abstract. The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. M c Intosh generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms.Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.

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Section: The M C Intosh Conditionmentioning

confidence: 91%

A generalisation of the form method for accretive forms and operators

Elst

Sauter

Vogt

2015

Journal of Functional Analysis

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“…Similar problems are investigated for the nodal count. It was asked a few years ago by Hoffmann-Ostenhof whether lim sup n→∞ ν n = ∞ holds for any manifold [59]. Following this, Ghosh, Reznikov and Sarnak proved that the number of nodal domains of Maass forms tends to infinity with the eigenvalue [32].…”

mentioning

confidence: 99%

Neumann Domains on Graphs and Manifolds

Alon¹,

Band²,

Bersudsky³

et al. 2018

Preprint

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The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in [2,6,7,46,62] and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.

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