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

Jussi Behrndt,Jonathan Rohleder

doi:10.1016/j.aim.2015.08.016

Jussi Behrndt, Jonathan Rohleder

Open Access

https://doi.org/10.1016/j.aim.2015.08.016

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Journal: Advances in Mathematics	Publication Date: Sep 8, 2015
Citations: 56	License type: cc-by

Affiliation: Graz University of Technology

Abstract

The spectrum of a selfadjoint second order elliptic differential operator in L2(Rn) 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.

Highlights

The Titchmarsh–Weyl function is an indispensable tool in direct and inverse spectral theory of ordinary differential operators and more general systems of ordinary differential equations; see the classical monographs [17, 55] and [11, 18, 27, 28, 29, 34, 38, 44, 51, 52] for a small selection of more recent contributions
One of the main objectives of this paper is to extend the classical spectral analysis of ordinary differential operators via the Titchmarsh–Weyl functions in (1.2) to the multidimensional setting
In the general setting of quasi boundary triples and their Weyl functions we show that a local simplicity condition on an open interval ∆ ⊂ R suffices to characterize the spectrum of A0 in ∆

Summary

Introduction

The Titchmarsh–Weyl function is an indispensable tool in direct and inverse spectral theory of ordinary differential operators and more general systems of ordinary differential equations; see the classical monographs [17, 55] and [11, 18, 27, 28, 29, 34, 38, 44, 51, 52] for a small selection of more recent contributions. The correspondence between the spectrum of the particular selfadjoint extension A0 := S∗ ↾ ker Γ0 and the limits of the Weyl function is not a special feature of the boundary triple for the above Sturm–Liouville equation It holds as soon as the symmetric restriction S (and, the boundary mappings Γ0 and Γ1) is chosen properly. Let S be a closed, densely defined, symmetric operator in H and let {G, Γ0, Γ1} be a quasi boundary triple for T = S∗ with A0 = T ↾ ker Γ0.

Spectral properties of selfadjoint operators and corresponding Weyl functions

Findings

Second order elliptic differential operators on Rn