On the essential spectrum of a class of singular matrix differential operators. II. Weyl's limit circles for the Hain–Lüst operator whenever quasi-regularity conditions are not satisfied

Abstract

The essential spectrum of the singular matrix differential operator of mixed order determined by the operator matrix

$$ \begin{pmatrix} -\dfrac{\mathrm{d}}{\mathrm{d} x}\rho(x) \dfrac{\mathrm{d}}{\mathrm{d} x}+q(x) & \dfrac{\mathrm{d}}{\mathrm{d} x}\dfrac{\beta(x)}{x} \\[12pt] -\dfrac{\beta(x)}{x}\dfrac{\mathrm{d}}{\mathrm{d} x} & \dfrac{m(x)}{x^2} \end{pmatrix} $$

is studied. Investigation of the essential spectrum of the corresponding self-adjoint operator is continued but now without assuming that the quasi-regularity conditions are satisfied. New conditions that guarantee that the operator is semi-bounded from below are derived. It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential operator is given by the range $\text{range}((m\rho-\beta^2)/\rho x^2)$ in the case where the quasi-regularity conditions are not satisfied.