Eigenfunction Expansions for Schrödinger Operators on Metric Graphs

“…(a) Under suitable assumptions it is possible to show a converse to the previous theorem i.e. every λ admitting an subexponentially bounded generalized eigenfunction must then belong to the spectrum of L. This type of result is known as Shnol theorem (see [8,14,34,35] for recent results of this type for operators arising from Dirichlet forms and further references).…”

Section: Metric Measure Spaces and Finer Growth Propertiesmentioning

confidence: 90%

Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces

Lenz

Teplyaev

2015

Trans. Amer. Math. Soc.

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Abstract. We consider an arbitrary selfadjoint operator in a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions, in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces consisting of general eigenfunctions. This automatically gives a Plancherel type formula. For suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions. For Laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the corresponding difference equation. Contents

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“…Note also that e −t(H0+V ) is not required to map L (4) The Laplacian on quantum or metric graphs is spatially locally compact under quite general assumptions, since D(H 0 ) is continuously embedded in L ∞ , see [8].…”

Section: Corollary 26 Assume That H 0 Is Spatially Locally Compact mentioning

confidence: 99%

Compactness of Schrödinger semigroups

Lenz

Stollmann

Wingert

2009

Mathematische Nachrichten

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Key words Compact resolvent, Schrödinger operators MSC (2000) 47B07, 35Q40, 47N50 To the memory of Erhard Schmidt, one of the founders of functional analysisThis paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of self-adjoint operators that are bounded below (on an L 2 -space). For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of the semigroup in terms of relative compactness of the operators of multiplication with characteristic functions of sublevel sets. In the context of Dirichlet forms, we can even characterize compactness of the semigroup for measure perturbations. Here, certain 'averages' of the measure outside of compact sets play a role.As an application we obtain compactness of semigroups for Schrödinger operators with potentials whose sublevel sets are thin at infinity.

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Eigenfunction Expansions for Schrödinger Operators on Metric Graphs

Abstract: We construct an expansion in generalized eigenfunctions for Schrö-dinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.

Cited by 17 publications

References 16 publications

Sch’nol’s theorem for strongly local forms

Sch’nol’s theorem for strongly local forms

Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces

Compactness of Schrödinger semigroups

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