Essential spectra of singular matrix differential operators of mixed order in the limit circle case

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