Spectral properties of general selfadjoint, even-order differential operators (Q2777502)

The singular even-order differential operator NEWLINE\[NEWLINE \ell(y)\equiv \frac 1{w(t)}\sum _{k=0}^n(-1)^k\left(p_k(t)y^{(k)} \right)^{(k)},\qquad t\in I=\langle a,\infty) , NEWLINE\]NEWLINE is studied in the paper. The author proves a necessary and sufficient condition under which the spectrum of any selfadjoint extension of the minimal differential operator generated by \(\ell \) in the weighted Hilbert space \(L^2_w\) is discrete and bounded from below. This is a generalization of a result concerning the one-term operator \(\ell(y)= \frac {(-1)^n}{w(t)}(p(t)y^{(n)})^{(n)}\) by \textit{O. Došlý} [Math. Nachr. 188, 49-68 (1997; Zbl 0889.34029)].NEWLINENEWLINENEWLINEThe results are illustrated using the special operator \(N(y)\equiv y''+\frac 1{4t^2}y\).