Some Remarks on the Krein-von Neumann Extension of Different Laplacians

Mugnolo, Delio

doi:10.1007/978-3-319-12145-1_5

Cited by 3 publications

(2 citation statements)

References 31 publications

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“…The Krein-von Neumann extension for a Laplacian on a metric graph has not been considered much in the literature so far; an attempt for a symmetric operator with vertex conditions different from the ones considered here was done in [38]. The Krein-von Neumann extension of our operator S is, like for the minimal Laplacian on a Euclidean domain, an operator with non-local vertex conditions.…”

Section: Introductionmentioning

confidence: 99%

The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

Muller

Rohleder

2021

Complex Anal. Oper. Theory

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The Krein–von Neumann extension is studied for Schrödinger operators on metric graphs. Among other things, its vertex conditions are expressed explicitly, and its relation to other self-adjoint vertex conditions (e.g. continuity-Kirchhoff) is explored. A variational characterisation for its positive eigenvalues is obtained. Based on this, the behaviour of its eigenvalues under perturbations of the metric graph is investigated, and so-called surgery principles are established. Moreover, isoperimetric eigenvalue inequalities are obtained.

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Section: Introductionmentioning

confidence: 99%

The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

Muller

Rohleder

2021

Complex Anal. Oper. Theory

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“…13.12] ∆ |D admits self-adjoint extensions. If H is a graph, then it has been proved in [11] that the usual quantum graph Laplacian -i.e., the second derivative with continuity and Kirchhoff-type boundary conditions in the nodes -agrees with the Friedrichs extension of ∆ |D . We regard the Friedrichs extension of ∆ |D as the canonical Laplacian for general hypergraphs, too.…”

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confidence: 99%

Laplacians on quantum hypergraphs

Mugnolo

2014

Proc Appl Math and Mech

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We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph. c 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim A hypergraph is a pair H := (V, E), where V is a set of nodes but, unlike in the more usual case of a graph, E is some set each of whose elements e -so-called hyperedges -is an unordered n e -tuple of elements of V, where each n e ∈ N may depend on e. Thus we obtain graphs as a special case of hypergraphs if n e ≡ 2. Conversely, a graph is canonically associated with H upon replacing each hyperedge e by a collection of 1 2 n e (n e − 1) edges that connect any two distinct elements of e: This graph is called the section of H in [1]. Observe that the connectivity of a hypergraph cannot be reconstructed from its section.If an orientation is assigned to each edge of a simple graph G := (V, E), i.e., if each edge is regarded as an ordered pair e ≡ (e init , e term ) ∈ V × V \ {(v, v) : v ∈ V}, then it is possible to define the incidence matrix I = (ι ve ) byotherwise, and the graph Laplacian by L := II T . Of course, L is a symmetric and positive semidefinite operator on the node space R V . The first attempts to develop a systematic theory of operators on R V -and in particular of graph Laplacians -date back to the 1970s: This spectral graph theory is nowadays rather mature, cf. [3,8]. One may want to introduce a hypergraph Laplacian as the graph Laplacian of the non-oriented hypergraph's section. This choice is made e.g. in [16] and a few subsequent papers on spectral clustering, but we argue that in this way the essential non-binary structure of a hypergraph is lost.An alternative approach seems to be more accurate and ultimately more appropriate for our purposes: Oriented hypergraphs have been introduced independently by many authors, see e.g. the historical remarks in [7], but it was only in the last few years that an algebraic hypergraph theory has been developed by their means. Indeed, upon regarding each hyperedge as an oriented pair e ≡ (e init , e term ) of disjoint subsets of V it is possible to introduce an incidence matrix I = (ι ve ) byotherwise.The algebraic properties of a hypergraph's incidence matrix are not as well understood as in the graph case. E.g., the combinatorial meaning of the rank and the co-rank of I seems to be unknown. Nevertheless, it is again possible to define a symmetric, positive semidefinite hypergraph Laplacian by L := II T . Its algebraic properties have been studied in [6] and more recently in [9,13,14], while here we briefly dip into the time-continuous evolution equationThe associated Cauchy problem is well-posed if the hypergraph is finite, but in general it is not associated with a Markov process, as one sees already in the simple case of a hypergraph consisting of three nodes and one hyperedge e with e int = {v 1 , v 2 } and e term = {v 3 }. In this case the incidence matrix, the hypergraph Laplacian, and t...

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Bi-Laplacians on graphs and networks

Gregorio

Mugnolo

2019

J. Evol. Equ.

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We study the differential operator A = d 4 dx 4 acting on a connected network G along with L 2 , the square of the discrete Laplacian acting on a connected discrete graph G. For both operators we discuss well-posedness of the associated linear parabolic problemsIn view of the well-known lack of parabolic maximum principle for all elliptic differential operators of order 2N for N > 1, our most surprising finding is that, after some transient time, the parabolic equations driven by −A may display Markovian features, depending on the imposed transmission conditions in the vertices.Analogous results seem to be unknown in the case of general domains and even bounded intervals. Our analysis is based on a detailed study of bi-harmonic functions complemented by simple combinatorial arguments. We elaborate on analogous issues for the discrete bi-Laplacian; a characterization of complete graphs in terms of the Markovian property of the semigroup generated by −L 2 is also presented.

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Some Remarks on the Krein-von Neumann Extension of Different Laplacians

Cited by 3 publications

References 31 publications

The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

Laplacians on quantum hypergraphs

Bi-Laplacians on graphs and networks

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