Boundary triplets and M‐functions for non‐selfadjoint operators, with applications to elliptic PDEs and block operator matrices

By Malcolm Brown, Marco Marletta, Serguei Naboko, Ian Wood. 2007

93 citations recorded by the source corpus.

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M‐function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M‐function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M‐function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M‐function corresponds to the Dirichlet to Neumann map.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5470f546c60067815ac0d0113a995af8c6c17097

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Citations in the corpus

A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains (2009, background): “d only very recently in the work of Amrein and Pearson [10], Behrndt and Langer [23], Brown, Grubb, and Wood [33], Brown, Hinchliﬀe, Marletta, Naboko, and Wood [34], Brown, Marletta, Naboko, and Wood [36], Gesztesy and Mitrea [56], [57], Grubb [67] (including a discussion of non-self-adjoint extensi”
Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (2012, background): “[Brown, Grubb and Wood, 2009; Brown et al., 2008; Grubb, 2009; Malamud, 2010; Posilicano and Raimondi, 2009] and is already essentially contained in the classical paper [Grubb, 1968].”
Robin-to-Robin Maps and Krein-Type Resolvent Formulas for Schr\ (2008, background): “ier papers [32] and [36]. For related literatureonWeyl–Titchmarshoperators,relevantin the context ofboundaryvalue spaces(boundary triples, etc.), we refer, for instance, to [2], [4], [11], [12], [16]–[20], [25]– [28], [31], [34], [37, Ch. 3], [39, Ch. 13], [54], [55], [59], [64], [65], [68]–[71], [7”
Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators (2012, background): “erators 2.1. Finite rank singular perturbations. We begin with the following deﬁnition, motivated by [6], where a more general case of closed linear relations was treated. We refer to[24], [5], [78], [19] and to books [35], [59], [53] for alternative treatments of singular perturbations in more gene”
Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schr\ (2008, background): “ier papers [43] and [46]. For related literatureonWeyl–Titchmarshoperators,relevantin the context ofboundaryvalue spaces(boundary triples, etc.), we refer, for instance, to [3], [5], [12], [13], [18]–[22], [32]– [35], [42], [44], [47, Ch. 3], [49, Ch. 13], [65], [66], [71], [80], [81], [84], [85], [”