Principal fundamental system of solutions, the Hartman-Wintner problem and correct solvability of the general Sturm-Liouville equation

Abstract

We study the problem of correct solvability in the space $L_p({\mathbb {R}}),$ $p\in [1,\infty )$ of the equation

$$\begin{aligned} -(r(x) y'(x))'+q(x)y(x)=f(x),\quad x\in {\mathbb {R}} \end{aligned}$$

under the conditions

$$\begin{aligned} r>0,\quad q\ge 0,\quad \frac{1}{r}\in L_1({\mathbb {R}}),\quad q\in L_1({\mathbb {R}}). \end{aligned}$$