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Spectral Analysis of Higher Order Differential Operators I: General Properties of the M ‐Function
Author(s) -
Remling Christian
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006474
Subject(s) - differential operator , mathematics , multiplicity (mathematics) , hilbert space , pure mathematics , spectral function , order (exchange) , function (biology) , differential (mechanical device) , differential equation , spectral properties , mathematical analysis , spectrum (functional analysis) , function space , spectral theorem , first order , operator theory , physics , quantum mechanics , finance , evolutionary biology , biology , thermodynamics , astrophysics , economics , condensed matter physics
As is well known, the classical Titchmarsh–Weyl m ‐function for second order differential operators admits a generalisation to considerably larger classes of differential equations. In the applications, however, the use of the general M ‐matrix often turns out to be rather cumbersome. The paper interprets the m ‐function in terms of Hilbert space notions, and shows that, in a sense that is made precise, the classical m ‐function can be recovered as a part of the more complicated one. Applications of this trick lead to results on spectral multiplicity and on stability properties of the spectrum.