A criterion for correct solvability of the Sturm-Liouville equation in the space $L_{p}(R)$

We consider an equation

where $f(x) \in L_{p}(R), p\in [1,\infty ] \left (\Vert f \Vert _{\infty } := C (R) \right )$, and $0 \le q(x)\in L_{1}^{\operatorname{loc}} (R).$By a solution of equation (1), we mean any function $y(x)$ such that $y(x), y'(x) \in AC^{\operatorname{loc}} (R),$and equality (1) holds almost everywhere on $R.$In this paper, we obtain a criterion for the correct solvability of (1) in $L_{p} (R)$, $p \in [1,\infty ].$