Weyl function and spectral properties of self-adjoint extensions

Brasche, Johannes F.; Malamud, M. M.; Neidhardt, Hagen

doi:10.1007/bf01255563

Cited by 84 publications

(112 citation statements)

References 19 publications

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“…The theory of boundary value spaces (also known as boundary triplets) associated with symmetric operators has its origins in the work of Kočubeȋ [24] and Gorbachuk and Gorbachuk [14] with developments from many authors, (see [6,25,27,28,35,37,39,41]). In this context, the theory of the Weyl-M -function was developed by Derkach and Malamud [9,10], where spectral properties of the operator were investigated via the M -function and Kreȋn-type resolvent formulae were established.…”

Section: Introductionmentioning

confidence: 99%

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Brown

Grubb

Wood

2009

Mathematische Nachrichten

140

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Key words Closed extension, M -function, abstract boundary spaces, boundary triplets, elliptic PDEs, pseudodifferential boundary operators, essential spectrum MSC (2000) 35J25, 35J30, 35J55, 35P05, 47A10, 47A11 Dedicated to the memory of Leonid R. VolevichIn this paper, we combine results on extensions of operators with recent results on the relation between the M -function and the spectrum, to examine the spectral behaviour of boundary value problems. M -functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M -function.

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Section: Introductionmentioning

confidence: 99%

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Brown

Grubb

Wood

2009

Mathematische Nachrichten

140

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“…We first consider the case 14) which illustrates the principal idea of the proof. To get started, we pick f, g ∈ L…”

Section: Krein-type Resolvent Formulasmentioning

confidence: 99%

Boundary Data Maps for Schrödinger Operators on a Compact Interval

Clark

Gesztesy

Mitrea

2010

Math. Model. Nat. Phenom.

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Abstract. We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0, R] with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context.Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrödinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to different (separated) boundary conditions, and a derivation of the Herglotz property of boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the special self-adjoint case.

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“…Algebraic multiplicities of eigenvalues in ρ(Q Σ + ⊕ Q Σ + ) can be found using Krein's resolvent formula (see [21,22] for a convenient abstract form), root subspaces for eigenvalues in ρ(Q Σ + ⊕ Q Σ + ) were found in [19]. Theorem 3.3 has some common points with [12], where the abstract Weyl function was used to find eigenvalues of a selfadjoint operator. But the approach of the present paper goes in the backward direction: we use the spectral measures dΣ ± and the functional model to find eigenvalues and root subspaces and then, using Lemma 3.2, return to the answer in the terms of the abstract Weyl functions given in Theorem 3.3.…”

Section: If λ ∈mentioning

confidence: 99%