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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Behrndt, Jussi¹  Gesztesy, Fritz²  Holden, Helge³  Nichols, Roger⁴ 

[1] Graz Univ Technol, Inst Numer Math, Steyrergasse 30, A-8010 Graz, Austria

[2] Univ Missouri, Dept Math, Columbia, MO 65211 USA

[3] Norwegian Univ Sci & Technol, Dept Math Sci, NO-7491 Trondheim, Norway

[4] Univ Tennessee, Dept Math, 415 EMCS Bldg,Dept 6956,615 McCallie Ave, Chattanooga, TN 37403 USA

关键词: Index computations for meromorphic operator-valued functions;    Dirichlet-to-Neumann maps;    Non-self-adjoint Schrodinger operators;    Boundary triples;    Weyl functions;    Donoghue-type M-functions;   

DOI  :  10.1016/j.jde.2016.05.033

来源: Elsevier

【 摘 要 】

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 Schrodinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh M-functions and Donoghue-type 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 Schrodinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl-Titchmarsh M-functions, respectively. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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