A criterion for compactness in \(L_p (\mathbb R)\) of the resolvent of the maximal general form Sturm-Liouville operator (Q2832274)

In this paper, the authors are concerned with the Sturm-Liouville operator NEWLINE\[NEWLINE {\mathcal L}_p y(x)= -(r(x) y'(x))' +q(x) y(x)NEWLINE\]NEWLINE acting on the Lebesgue spaces \(L_p({\mathbb R})\), \(1<p<\infty\). Here, \(r(x)>0\) and \(q(x)\geq 0\) are such that the problem NEWLINE\[NEWLINE {\mathcal L}_p y=f NEWLINE\]NEWLINE is correctly solvable, i.e., \({\mathcal L}_p^{-1}: L_p({\mathbb R}) \to L_p({\mathbb R})\) is well-defined and bounded. In Theorem 3.1, the authors give a necessary and sufficient condition on \(r, \;q\) so that \({\mathcal L}_p^{-1}: L_p({\mathbb R}) \to L_p({\mathbb R})\) is also a compact linear operator. The condition is involved of some extension of the Steklov average values. Moreover, in Theorem 3.2, an estimate on the minimal eigenvalue of \({\mathcal L}_p\) is given. All of these results can be explicitly expressed in terminology of the coefficients \(r(x), \;q(x)\).NEWLINENEWLINEThe present paper is based on many results from the authors [Proc. Am. Math. Soc. 127, No. 5, 1413--1426 (1999; Zbl 0918.34032); J. Lond. Math. Soc., II. Ser. 80, No. 1, 99--120 (2009; Zbl 1188.34036)] and \textit{P. A. Zharov} [Proc. Steklov Inst. Math. 194, 101--114 (1993; Zbl 0811.26007); translation from Tr. Mat. Steklova 194, 97--110 (1992)].