Spectral theory of ordinary linear differential operators (Q2734848)

The formally selfadjoint differential expression NEWLINE\[NEWLINE\tau y=w^{-1}[(ry'')''-(py')'+qy]NEWLINE\]NEWLINE is considered on \([0,\infty)\), where \(p\), \(q\), \(r\), \(w\) are real-valued, \(r,w>0\), and \(r^{-1},p, q, w\in L_{1,loc}([0,\infty))\). The corresponding minimal operator has the defect indices \((m,m)\) with \(m=2\), \(3\), or \(4\). It is investigated how these defect indices and the spectra of selfadjoint realizations depend on the asymptotic behavior of the coefficients. A crucial role is played by the \(M\)-matrix, the analogon of the Titchmarsh-Weyl coefficient in the present case. According to \textit{H. Behncke} [Proc. Am. Math. Soc. 111, No. 2, 373-384 (1991; Zbl 0726.34072) and Manuscr. Math. 71, No. 2, 163-181 (1991; Zbl 0726.34073)], the coefficient functions can be written as sums of smooth (s), conditionally integrable (c), and integrable (i) functions. Introducing the functions \(\widetilde p=p_s/(rw)^{\frac 12}\), \(\widetilde q=q_s/w\), properties of the spectrum such as location of the continuous spectrum, point spectrum and limit points of the point spectrum are investigated in the following cases: \(\widetilde p\), \(\widetilde q\) bounded; \(\widetilde q\to \infty \); \(\widetilde q\to-\infty \), \(\widetilde p^2=O(\widetilde q)\); \(\widetilde p\to\pm\infty \), \(\widetilde q=o(\widetilde p^2)\). Finally, it is shown that the matrix version of the Kummer-Liouville transform is useful to obtain information on the spectrum.