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Essential spectra of singular matrix differential operators of mixed order in the limit circle case
Author(s) -
Qi Jiangang,
Chen Shaozhu
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810062
Subject(s) - mathematics , differential operator , hamiltonian (control theory) , limit (mathematics) , spectrum (functional analysis) , operator (biology) , pure mathematics , integrable system , essential spectrum , mathematical analysis , matrix (chemical analysis) , spectral theory of ordinary differential equations , singular value , order (exchange) , mathematical physics , eigenvalues and eigenvectors , quantum mechanics , quasinormal operator , physics , finite rank operator , materials science , repressor , banach space , chemistry , composite material , mathematical optimization , biochemistry , transcription factor , finance , economics , gene
Abstract The present paper is concerned with the essential spectrum of the singular matrix differential operator of mixed order\documentclass{article}\usepackage{amsmath}\begin{document}\pagestyle{empty}$$ \begin{eqnarray*} T=\begin{pmatrix} -Dp(t)D+q(t) & -D h(t)\\[6pt] \bar{h}(t) D & d(t) \end{pmatrix} \end{eqnarray*} $$\end{document} over the interval ( a , b ), where D = d / dt . It is shown that if the coefficients are locally integrable functions and T is in the limit circle case, then the essential spectrum of T is given by the essential range of ${{p(t)d(t)-\vert h(t)\vert^2}\over{p(t)}} $\end{document} . The main idea is to transform the original spectral problem into a spectral problem of a singular Hamiltonian differential operator so that the classical Titchmarsh‐Weyl theory can apply. The approach used here can be applied to many other cases. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim