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

详细信息    查看全文

• 作者：Jussi Behrndt^a ; ^{behrndt@tugraz.at" class="auth_mail" title="E-mail the corresponding author} ; Fritz Gesztesy^b ; ^{gesztesyf@missouri.edu" class="auth_mail" title="E-mail the corresponding author} ; Helge Holden^c ; ^{holden@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Roger Nichols^d ; ^{Roger-Nichols@utc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 47A53 ; 47A56 ; secondary ; 47A10 ; 47B07
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 September 2016
• 年：2016
• 卷：261
• 期：6
• 页码：3551-3587
• 全文大小：537 K

文摘

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 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 Schrödinger 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.