Spectral analysis of selfadjoint elliptic differential operators, Dirichlet-to-Neumann maps, and abstract Weyl functions

Abstract

The spectrum of a selfadjoint second order elliptic differential operator in L 2 (R n ) is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional Glazman decomposition and correspond to an interior and an exterior boundary value problem. This leads to PDE analogs of renowned facts in spectral theory of ODEs. The main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding Weyl functions, and are applied to the PDE setting afterwards.

Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I

“…4 "Very weak solutions" in the terminology of [52]. 5 These left-inverse operator is well-defined on ran Π stiff(soft) . Note that neither its closedness nor closability is assumed.…”

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

“…In addition, we refer the reader to [1,2,[9][10][11][12][13][14][15][16]18,[20][21][22][23]37,38,[48][49][50][51][52][53][54][55][56][57][58][59][60], for more details, applications, and references on boundary triples and their Weyl-Titchmarsh functions.…”

“…In contrast to the index formula (1.3) in Section 3, here the values of the WeylTitchmarsh function M(·) are bounded operators, but the parameters 1 and 2 are in general unbounded, closed operators. However, the strategy and the difficulties in the proof of the index formula in Theorem 4.3 are similar to those in Section 3: One first has to verify that the generalized index is well-defined for the functions 1 − M(·) and 2 − M(·) (again the key difficulty is to show that the inverses ( 1 − M(·)) −1 and ( 2 − M(·)) −1 are finitely meromorphic at a discrete eigenvalue of B 1 and B 2 , respectively) and then a Krein-type resolvent formula (see, e.g., [1,2,9,11,12,[14][15][16][21][22][23][24][27][28][29][30]32,38,[43][44][45][46][47]61], and the references cited therein) yields the index formula (1.6).…”

Self-adjoint indefinite Laplacians

“…We hope that, in view of the recent progress in the theory of self-adjoint extensions of partial differential operators, see e.g. [5,6,15], such a direct reformulation could be a starting point for a further advance in the study of indefinite operators.…”