On the index of meromorphic operator-valued functions and some applications

Behrndt, Jussi; Gesztesy, Fritz; Holden, Helge; Nichols, Roger

doi:10.4171/175-1/5

Cited by 6 publications

(5 citation statements)

References 35 publications

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“…Proposition 18. (see for instance Gohberg and Leiterer 31 and Behrndt et al 32 ). Let ( ) ∶ X → X be an analytic Fredholm family in , and there exists a point 0 ∈  such that ( 0 ) ∶ X → X is an invertible operator.…”

Section: Reduction Of Spectral Problem (53) To a Pseudodifferential Equation On σmentioning

confidence: 96%

“…The first equation in system (60) yields that v + = ( )v − . Hence, the second equation in system (60) takes the form We recall some definition and results concerning analytic families of Fredholm operators (see for instance Gohberg and Leiterer 31 and Behrndt et al 32 ). Definition 17.…”

Section: Reduction Of Spectral Problem (53) To a Pseudodifferential Equation On σmentioning

confidence: 99%

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Magnetic Schrödinger operators with delta‐type potentials

Rabinovich

2020

Math Methods in App Sciences

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We consider the magnetic anisotropic Schrödinger operator on where is the vector potential of the magnetic field and W(x) is the scalar potential of the electric field. We assume that aj and ϱij are real‐valued functions belonging to the space of bounded with first derivatives on functions, whereas is a complex‐valued electric potential. Let Σ be a C2‐hypersurface in dividing on two open domains Ω± with common boundary Σ. We assume that Σ is a closed C2‐hypersurface or unbounded hypersurface of bounded geometry. We consider the magnetic Schrödinger operator with singular potentials Ws with supports on Σ. We associate with an unbounded operator in generated by Hϱ,a, W with domain in H2(Ω+) ⊕ H2(Ω−) consisting of functions satisfying interaction conditions on Σ. We study the self‐adjointness of the operator and its Fredholm properties. Moreover, we consider the spectral problem and reduce this problem to a spectral problem for some boundary pseudodifferential operators on Σ. We describe the domains in where problem (2) has the discrete spectrum.

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Section: Reduction Of Spectral Problem (53) To a Pseudodifferential Equation On σmentioning

confidence: 96%

Section: Reduction Of Spectral Problem (53) To a Pseudodifferential Equation On σmentioning

confidence: 99%

Magnetic Schrödinger operators with delta‐type potentials

Rabinovich

2020

Math Methods in App Sciences

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“…This leads to the following definition of a generalized index of M(·), which extends the notion of an index for finitely meromorphic B(H)-valued functions employed in [36] and, for instance, in [33,35] (cf. [7,Definition 4.2]). …”

Section: Example 22mentioning

confidence: 99%

Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions

Behrndt

Gesztesy

Holden

et al. 2016

Journal of Differential Equations

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We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh M-functions and Donoghuetype M-functions corresponding to closed extensions of symmetric operators belong to it.The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schrödinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the ✩ JID:YJDEQ AID:8393 /FLA [m1+; v1.232; Prn:20/06/2016; 10:29] P.2 (1-37) 2 J. Behrndt et al. / J. Differential Equations ••• (••••) •••-••• corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl-Titchmarsh M-functions, respectively.

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“…Malamud and V.I. Mogilevskii in [48,49,50,53]; see also [10,40,41]. For the special case of symmetric operators and the spectral analysis of their selfadjoint extensions the boundary triple technique is nowadays very well established [11,23,24,27,31,36,39,42,60] and has been applied and extended in various directions.…”

Section: Introductionmentioning

confidence: 99%

Boundary Value Problems, Weyl Functions, and Differential Operators

Behrndt

Hassi

Snoo

2020

Monographs in Mathematics

134

227

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The foundations of this outstanding book series were laid in 1944. Until the end of the 1970s, a total of 77 volumes appeared, including works of such distinguished mathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. The series came to its name and present appearance in the 1980s. In keeping its well-established tradition, only monographs of excellent quality are published in this collection. Comprehensive, in-depth treatments of areas of current interest are presented to a readership ranging from graduate students to professional mathematicians. Concrete examples and applications both within and beyond the immediate domain of mathematics illustrate the import and consequences of the theory under discussion.

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On the index of meromorphic operator-valued functions and some applications

Cited by 6 publications

References 35 publications

Magnetic Schrödinger operators with delta‐type potentials

Magnetic Schrödinger operators with delta‐type potentials

Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions

Boundary Value Problems, Weyl Functions, and Differential Operators

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